This is a review of an unpublished decision of the court of appeals affirming an order of the circuit court for Kenosha county, Judge William U. Zievers, granting the defendants’ motions to suppress evidence obtained as a result of the execution of a search warrant.
The issue before us concerns the circuit court’s conclusion that the affidavit contains false statements made knowingly and intentionally or with a reckless *455disregard for the truth. Based upon these conclusions, the circuit court determined that the affidavit, absent those statements, failed to state probable cause. The circuit court, therefore, suppressed the evidence which was obtained as a result of the invalid search warrant. Because we conclude, under the methodology outlined in Franks v. Delaware, 438 U.S. 154 (1978), that the circuit court erroneously determined that the statements in the affidavit were either intentionally false or made with reckless disregard for the truth, and because we find that the affidavit does state probable cause, we uphold the validity of the search warrant. We, therefore, vacate the circuit court’s order to suppress the evidence obtained as a result of the execution of the search warrant.
The facts in these cases are as follows: Paul W. Anderson (defendant) and Brian J. Hedges (defendant) were each charged separately with possession of marijuana with intent to deliver in violation of secs. 161.41(1m), 161.14(4)(t), and 939.05, Stats. (1983-84). The charges were based upon evidence seized in a search, pursuant to a search warrant, of a residence located at 1039 Sheridan Road, Township of Somers, Kenosha county, Wisconsin. Circuit Judge David M. Bastían had issued the warrant based on the following affidavit of Officer Gary R. Smith of the Kenosha Police Department.
"That your affiant is an officer with the City of Kenosha Police Department. Has knowledge of the here-in alledged (sic) facts from personal observation. In that on Sept. 10, 1984, at 10:45 a.m. your affiant was present at the parking lot of the Surfside Bowling Lanes along with Donald J. Kisielewski. Your affiant observed Mr. Kisielewski approach the above-described (sic) residence for the *456purpose of obtaining marihuana which purpose had been established during their conversation immediately prior to Mr. Kisielewski’s approach to said residence. Your affiant observed Mr. Kisielew-ski return at which time Mr. Kisielewski stated 'He’s out. He’ll be back shortly.’ Your affiant and Mr. Kisielewski waited approximately fifteen minutes at which time your affiant observed Mr. Kisielewski return to the above-described premises. Approximately one minute later, Mr. Kisie-lewski returned and stated to your affiant that he would 'get a sample of what your affiant was buying.’ Your affiant then observed Mr. Kisielew-ski return to the above-named residence and approximately five minutes returned to your affi-ant in said parking lot with a small plastic baggie containing a green leafy substance field tested by Detective Bernard Fapso which test did prove positive for the presence of marihuana. Mr. Kisie-lewski stated that the occupant of said residence had in his possession the three pounds of marihuana which your affiant had arranged to purchase from which three pounds the above-mentioned sample was taken. Your affiant further discussed with Mr. Kisielewski arrangements for the delivery of three pounds of marihuana being completed in the evening on Sept. 10, 1984. Your affiant further observed Mr. Kisielewski state there were five pounds of marihuana potentially available for your affiant’s purchase. Your affiant has knowledge as to the reliability of Mr. Kisielewski from two prior occasions when he provided your affiant with accurate information.”
Following a preliminary examination and bind-over, Anderson and Hedges filed separate motions to suppress all evidence seized during the search. Both motions raised two primary arguments. The defend*457ants first argued that the affidavit was insufficient to state probable cause because it failed to establish the reliability of Donald J. Kisielewski, the person from whom Officer Smith obtained his information about the location of the marijuana. Second, the defendants argued that, even if Kisielewski’s reliability was sufficiently shown, the affidavit nevertheless failed to state probable cause because it contained false statements made knowingly and intentionally or with reckless disregard for the truth and that such statements were necessary to state probable cause.1 Accordingly, the defendants requested an order suppressing the use of any evidence from, and all fruits of, the search conducted at 1309 Sheridan Road, Town of Somers, Wisconsin.
A consolidated hearing on both motions to suppress was held before Judge Zievers. The evidence introduced at the hearing consisted of the testimony of Officer Smith and a stipulation by the parties as to what the testimony of Judge Bastian would have been had he testified.
At the hearing, Officer Smith (Smith) testified that he had been a police officer since November, 1978, and that he had been assigned to narcotics for the year and a half immediately preceding the issuance of the warrant. Smith further testified to the following series of events leading up to the execution of the search warrant. On September 4, 1984, Smith, acting in his undercover capacity, purchased a quarter ounce of marijuana from Kisielewski. At the time of the purchase, Smith observed additional quantities of marijuana in Kisielewski’s residence.
*458On September 6, 1984, Smith obtained additional marijuana from Kisielewski. This second acquisition of marijuana was obtained without charge, leading up to a three-pound buy that Smith was supposed to make. Kisielewski advised Smith that Kisielewski’s source was in the north part of Kenosha county and that the three-pound purchase would come from Kisielewski’s source. At this second meeting, Smith again observed additional quantities of marijuana in Kisielewski’s apartment.
On September 10, 1984, Smith met Kisielewski and drove with him to the Surfside Bowling Lanes parking lot on Sheridan Road. While enroute, Kisie-lewski told Smith they were going to Kisielewski’s source on Sheridan Road.2 Once in the parking lot, Kisielewski parked the car at the south end of the lot facing south. Kisielewski then told Smith that, because no strangers were allowed in the house, Smith would have to wait in the car while Kisielewski went to the house. At that time Kisielewski indicated that the house involved was the one referred to in the search warrant. The house at 1039 Sheridan Road is the only residence in the vicinity of where the car was parked.
Smith then observed Kisielewski walk around the east section of the house. Because Smith lost sight of Kisielewski when he walked west along the house, Smith did not see Kisielewski enter the house. Ap*459proximately one minute later, Kisielewski returned from the area of the house and stated that "he” was not home. Smith and Kisielewski then went into the bowling alley. Sometime later an unidentified individual said "Don [Kisielewski], if you’re looking for Brian he’s home now.” After exiting the bar and returning, Kisielewski advised Smith that Brian was home and directed Smith to wait in the car.
Smith then observed Kisielewski walk around the east section of the house. Smith again did not see Kisielewski enter the house. Approximately three minutes later, Kisielewski returned with a small plastic bag. Kisielewski told Smith that Brian had the same stuff in the house and that this was a sample of what he was going to buy. Although Smith had sufficient money with him, the drug transaction did not occur. Kisielewski did not explain why the buy did not occur; he only stated that it would be put off until that night. Kisielewski and Smith then proceeded back to the apartment.
Smith further testified that, because of the amounts of marijuana in Kisielewski’s possession, Smith concluded that Kisielewski was more a dealer than a user. Although Smith testified that reliability of an informant is important and that he had never previously used evidence of criminal conduct as providing reliability, Smith considered Kisielewski to be a good contact and a reliable person. According to Smith, Kisielewski’s conduct with Smith is what led him to believe that Kisielewski was reliable.
On the afternoon of September 10, 1984, Smith and Assistant District Attorney J. Patrick O’Neill prepared and submitted Smith’s affidavit in support of the request for a search warrant. According to a stipulation entered into by the parties, "Judge Bastian *460examined the affidavit, read the entire affidavit and then indicated as to a concern that he had regarding the basis for the reliability of Mr. Kisielewski and that it was then at his insistence that [an] additional handwritten paragraph ... was added describing the basis for relying on Mr. Kisielewski’s information.” In response to Judge Bastian’s concerns, Assistant District Attorney O’Neill added the following additional lines at the bottom of the affidavit which Smith initialed. "Your affiant has knowledge as to the reliability of Mr. Kisielewski from two prior occasions when he provided your affiant with accurate information.” Judge Bastían then signed the search warrant.
The circuit court, in concluding that the evidence seized pursuant to the search warrant had to be suppressed, ruled that "the representations in the affidavit are, number one, a deliberate disregard for the truth in terms of representing Mr. Kisielewski to be a reliable informant, and at best a reckless disregard for the truth in the balance of its allegations about observing Mr. Kisielewski’s activities as they relate to the house in question.” According to the court, the law does not contemplate that the reliability of an informant will be sustained by participating in two controlled sales and purchases of drugs in harmony with the parties’ expectations. Thus, Smith’s statement that he had reliable information was a wanton disregard for the truth. Furthermore, the court noted that Smith’s assertion that he observed Kisielewski "return to” the house was, in light of the fact that Smith never saw Kisielewski enter the house, a reckless disregard for the truth. The court, therefore, concluded under Franks v. Delaware, 438 U.S. 154 (1978), that the evidence seized pursuant to the search warrant had to be suppressed.
*461The court of appeals affirmed the order of the circuit court. According to the court of appeals, the circuit court’s findings — that crucial statements in the affidavit were made falsely or with a reckless disregard for the truth — were not clearly erroneous. Moreover, the court of appeals agreed with the conclusion of the circuit court that, absent these statements, the affidavit did not state probable cause. On November 4, 1986, this court granted the state’s petition for consolidated review.
The challenges brought in these cases are based on the protections afforded by Article I, Section 11 of the Wisconsin Constitution,3 and the fourth amendment to the United States Constitution4 as applied to the states through the fourteenth amendment. We are bound by the United States Supreme Court’s interpretation of the fourth amendment to the United States Constitution. Furthermore, this court has previously recognized that Article I, Section 11 of the Wisconsin Constitution is substantially equivalent to the fourth amendment to the United States Constitution, and *462thus the legal analysis applicable to the fourth amendment is generally applicable to Article I, Section 11. See State v. Paszek, 50 Wis. 2d 619, 624, 184 N.W.2d 836 (1971); State v. Fry, 131 Wis. 2d 153, 172, 388 N.W.2d 565 (1986).
The issues in these cases reach us as a result of a challenge to the truthfulness of statements made in an affidavit in support of a search warrant, as well as the sufficiency of the affidavit. When challenging the veracity of statements in support of a search warrant, the defendant must first make a substantial preliminary showing that a false statement knowingly and intentionally, or with reckless disregard for the truth, was included by the affiant in the warrant affidavit and that the allegedly false statement is necessary to the finding of probable cause. Franks v. Delaware, 438 U.S. at 155-56.
To make a substantial preliminary showing "[t]here must be allegations of deliberate falsehood or of reckless disregard for the truth, and those allegations must be accompanied by an offer of proof. They should point out specifically the portion of the warrant affidavit that is claimed to be false; and they should be accompanied by a statement of supporting reasons.” Id. at 171. If the court concludes that the defendant has made the substantial preliminary showing, then the defendant is entitled to a hearing at which the defendant must then prove, by a preponderance of the evidence, that the challenged statement is false, that it was made intentionally or with reckless disregard for the truth, and that absent the challenged statement the affidavit does not provide probable cause. Id. at 156. If the defendant meets this *463burden, the fourth amendment requires that the search warrant be voided and the evidence discovered pursuant to the warrant be suppressed. Id.
Thus, when a challenge is made to a search warrant on the grounds that the affidavit contains false statements, the defendant is entitled to a Franks hearing,5 at which evidence is presented to determine the veracity of the challenged statements. A presumption of validity attends the affidavit. Id. at 171. To overcome this presumption, the defendant must first prove that the challenged statement is false, and second, that the affiant intentionally made the false statement or that the statement was made with a reckless disregard for the truth. Proof that the challenged statements were made innocently or negligently is insufficient to have the challenged statement removed from the affidavit. As noted earlier, both the proof that the statement is false and that it was made either intentionally, or with a reckless disregard for the truth, must be shown by a preponderance of the evidence.
Although Franks did not define "reckless disregard for the truth,” cases applying Franks have held that, to prove reckless disregard for the truth, the defendant must prove that the affiant in fact entertained serious doubts as to the truth of the allegations or had obvious reasons to doubt the veracity of the allegations. United States v. Williams, 737 F.2d 594, *464602 (7th Cir. 1984); United States v. Pritchard, 745 F.2d 1112, 1116 (7th Cir. 1984); United States v. Davis, 617 F.2d 677, 694 (D.C. Cir. 1979). Because the defendant must show either intent or reckless disregard, a Franks hearing, by necessity, focuses on the state of mind of the affiant.6
If the challenged statements are proven to be false and either made recklessly or intentionally, then the challenged statements are excised from the affidavit. The affidavit is then examined to determine whether, with the statements excised, the affidavit provides probable cause for a search warrant. If the affidavit fails to state probable cause, the warrant is voided and the evidence seized pursuant to the warrant must be suppressed. If with the challenged statements excised the warrant still states probable cause, the warrant is upheld and the evidence is admissible.
In the present cases, the state did not appeal the lower court’s determination that the defendants were entitled to a Franks hearing. We are thus only concerned with whether the defendants have shown, by a preponderance of the evidence, that false statements were made either intentionally or with a reckless disregard for the truth.7 Because we conclude that there is insufficient evidence to show that the *465challenged statements were either intentionally untruthful or made with reckless disregard for the truth, we hold that the challenged statements remain in the affidavit.
The first challenged statement is the final sentence in the affidavit in which the affiant stated: "Your affiant has knowledge as to the reliability of Mr. Kisielewski from two prior occasions when he provided your affiant with accurate information.” The circuit court determined that this statement was false, concluding that participation in two prior drug transactions, in harmony with prior arrangements, could not constitute receipt of reliable information. However, it is apparent from the testimony that the completion of two drug transactions, in accordance with prior arrangements, can support reliability and that the two transactions did, in fact, lead Smith to conclude that Kisielewski was reliable.
It is clear from the evidence introduced at the hearing that the affiant (Smith) had on two prior occasions arranged and consumated drug transactions with Kisielewski. At the hearing it was stated that Kisielewski represented to Smith that he could provide marijuana. Smith then twice obtained what was alleged to be and appeared to be marijuana from Kisielewski, and both times tests confirmed that it was marijuana. We see no significant distinction between an individual who states that he or she can provide marijuana and then does, and a situation in which an individual states that a third party will be involved in a drug transaction which subsequently occurs. In each instance, the individual has provided accurate information. A drug transaction has occurred as predicted. The individual’s participation in the transaction does not negate the fact that he or she *466declared, "I will provide marijuana,” and the declaration was accurate — he or she provided marijuana.
Moreover, it is reasonable for an undercover police officer to consider a drug supplier to be reliable because the drug supplier has twice provided what was said to be provided. Smith testified at the hearing that he considered Kisielewski to be reliable; Kisielew-ski did not lie to him, he did not send him on wild goose chases, and he did not lead Smith to believe that he could deliver something that he could not provide. It is reasonable in the context of drug transactions to determine that a dealer is reliable if he or she consumates deals in accordance with a prearranged agreement. Furthermore, the burden in a Franks hearing is not on the affiant; it is on the defendant.
The defendants in these cases presented no evidence that Smith believed or knew that his statement was untrue. Instead, the defendants appear to rely on the theory that past drug transactions cannot be used as a basis for determining reliability. Although the circuit court appeared to agree with this theory,8 we conclude, based on our discussion above, that the defendants have not proven that Smith’s statement was an intentional lie or one that was made with a reckless disregard for the truth. We note that accurate information comes in many different forms. To limit what is accurate information, as the circuit court did in the present cases, unnecessarily restricts a police *467officer’s reliance upon and use of statements of an individual.
We next turn to the question of whether Smith’s assertion that he observed Kisielewski twice "return to” the premises constituted an intentional lie or a reckless disregard for the truth. According to the circuit court, this language was a reckless disregard for the truth in light of the testimony which indicated that Smith never saw Kisielewski enter the premises. Apparently, the circuit court felt that the wording used by Smith implied that Smith had, in fact, observed Kisielewski enter the house, when in fact he had not, and that such an implication amounted to a reckless disregard of the truth. We disagree.
First, we note that Smith did not assert in the affidavit that he saw Kisielewski enter the house. Although Smith failed to state that he did not observe Kisielewski enter the house, this omission does not constitute a reckless disregard for the truth, let alone an intentional lie. Smith did not affirmatively state one way or the other whether he saw Kisielewski enter the house. We conclude that it is erroneous to label this omission a reckless disregard for the truth.
Having determined that the defendants failed to prove by a preponderance of the evidence that the challenged statements were either an intentional lie or made with a reckless disregard for the truth, we conclude that the circuit court erroneously excised the challenged statements from the affidavit.
The remaining issue is whether the affidavit, including the challenged statements, states probable *468cause to issue a search warrant.9 The existence of probable cause is determined by applying the "totality-of-the-circumstances” test adopted in Illinois v. Gates, 462 U.S. 213, reh’g denied, 463 U.S. 1237 (1983). In Gates the United States Supreme Court ruled that "[t]he task of the issuing magistrate is simply to make a practical commonsense decision whether, given all the circumstances set forth in the affidavit before him, including the 'veracity’ and 'basis of knowledge’ of persons supplying hearsay information, there is a fair probability that contraband or evidence of a crime will be found in a particular place. And the duty of a reviewing court is simply to ensure that the magistrate had a 'substantial basis for ... concluding]’ that probable cause existed.” Id. at 238-39; State v. Wheaton, 114 Wis. 2d 346, 352, 338 N.W.2d 322 (1983), overruled in part, State v. Pham, 137 Wis. 2d 31, 37, 403 N.W.2d 35 (1987).
In adopting the totality-of-the-circumstances test, the Court abandoned the two-pronged test established in Aguilar v. Texas, 378 U.S. 108 (1964), and Spinelli v. United States, 393 U.S. 410 (1969).10 According to the *469Court, the two-pronged test had resulted in an overly technical analysis of informants’ tips which erroneously resulted in each element becoming an entirely separate and independent requirement. Gates, 462 U.S. at 230, 234. Instead of a rigid requirement that specific tests be met, the Court adopted a more flexible, common-sense analysis designed to recognize that probable cause is concerned with probabilities, not hard certainties. However, the Court noted that the two-pronged test is not entirely abandoned. Instead, it is better understood as "relevant considerations in the totality-of-the-circumstances analysis ...: a deficiency in one may be compensated for, in determining the overall reliability of a tip, by a strong showing as to the other, or by some other indicia of reliability.” Id. at 233.
In reviewing the sufficiency of an affidavit, great deference must be paid to a magistrate’s determination of probable cause. So long as the magistrate had a substantial basis for concluding that a search would uncover evidence of wrongdoing, we uphold the determination of the magistrate. Id. at 236.
A number of factors convince us that the affidavit, under the totality-of-the-circumstances test, provided Judge Bastian with a substantial basis for concluding that the house at 1039 Sheridan Road contained controlled substances. First, Smith asserts in the affidavit that he saw Kisielewski several times approach and then return from the target house. In addition, Smith notes that, after the last trip toward the house, Kisielewski returned with a leafy, green substance which was later determined to be marijuana. Furthermore, the affidavit asserts that Kisielew-ski told Smith that the marijuana given to Smith was *470a sample of the three pounds of marijuana which the occupant of the house had in his possession.
When someone approaches a house, returns with marijuana, and states that the marijuana is part of a larger supply in the possession of the occupant of the house, a reasonable inference is created that the house is the source of the marijuana. These reasonable inferences arise as a result of Smith's first-hand observation of Kisielewski’s conduct in conjunction with Kisielewski’s statements regarding the source of the marijuana. Thus, before probable cause would exist to believe that the house contained marijuana, there must exist a basis for a judge to rely on Kisielewski’s out-of-court statements.
Based on the facts asserted in the affidavit, we conclude that there are creditable reasons supporting Judge Bastian’s decision to issue the search warrant. First is Smith’s assertion that he has knowledge of Kisielewski’s reliability from two prior occasions when Kisielewski provided Smith with accurate information. Second, the fact that Kisielewski’s actions were against his penal interest acts to enhance his reliability. Numerous cases recognize that the credibility of an informant, for the purpose of finding probable cause, is established by the fact that his or her statement is against his or her penal interest. E.g., State ex rel. Bena v. Hon. John J. Crosetto, 73 Wis. 2d 261, 266-67, 243 N.W.2d 442 (1976); Laster v. State, 60 Wis. 2d 525, 535, 211 N.W.2d 13 (1973); United States v. Harris, 403 U.S. 573, 583 (1971). In Harris the United States Supreme Court stated:
"Common sense ... would induce a prudent and disinterested observer to credit these state*471ments. People do not lightly admit a crime and place critical evidence in the hands of the police in the form of their own admissions. Admissions of crime, like admissions against proprietary interests, carry their own indicia of credibility — sufficient at least to support a finding of probable cause to search.” 403 U.S. at 583.
Furthermore, we agree with the conclusion reached by other courts that statements against penal interests may be used to establish reliability even when the declarant is not aware that the listener is a police officer. See State v. Collins, 56 N.C. App. 352, 356, 289 S.E.2d 37, 40 (1982); State v. Lair, 95 Wash. 2d 706, 710-11, 630 P.2d 427, 430 (1981); State v. Wiberg, 296 N.W.2d 388, 396 (Minn. 1980); See also 1 W. LaFave, Search and Seizure, sec. 3.3(c) at 652 (1987) (admissions against penal interest can be used to establish reliability even if statement is made to someone other than a police officer). Kisielewski’s delivery of the sample violated sec. 161.41, Stats (1983-84), and thus constituted an admission against his penal interests. Moreover, the delivery by Kisie-lewski was directly related to the circumstances described in the affidavit.
We conclude that the affidavit, read as a whole, provided a substantial basis for Judge Bastían to determine that Kisielewski was reliable. Because we reach this conclusion on the basis of the affidavit as a whole, we do not reach the issue of whether either Smith’s statements in support of Kisielewski’s reliability or Kisielewski’s admission against penal interest would, on its own, provide such a basis.
*472Therefore, based on the totality of the circumstances, including Kisielewski’s statement against penal interest, Kisielewski’s providing of accurate information in the past, the delivery of a sample which was later determined to be marijuana, and Kisielewski’s indication that the marijuana was in the target house, we hold that Judge Bastían had a substantial basis for concluding that marijuana was located at 1039 Sheridan Road. We therefore affirm Judge Bastian’s determination that the affidavit stated probable cause to search the residence located at 1039 Sheridan Road, Township of Somers, Kenosha County, Wisconsin.
By the Court — The decision of the court of appeals is reversed, and the circuit court’s order to suppress the evidence seized pursuant to the search warrant is vacated.