From industry statistics, a credit card company knows that 0.8 of its potential card holders are good credit risks, and 0.2 are bad credit risks. The company uses discriminant analysis to screen credit card applicants and determine which ones should receive credit cards. The company awards credit cards to 70% of those who apply. The company has found that of those awarded credit cards, 95% turn out to be good credit risks. What is the probability that an applicant who is a bad credit risk will be denied a credit card?

To answer this question, we can use the concept of conditional probability.

Let's define the following events:
A: Applicant is a good credit risk
B: Applicant is a bad credit risk
C: Applicant is awarded a credit card

According to the given information, we have the following probabilities:
P(A) = 0.8 (probability of an applicant being a good credit risk)
P(B) = 0.2 (probability of an applicant being a bad credit risk)
P(C|A) = 0.95 (probability of being awarded a credit card given that the applicant is a good credit risk)
P(C|B) = ? (probability of being awarded a credit card given that the applicant is a bad credit risk)

To find the probability that an applicant who is a bad credit risk will be denied a credit card, we need to find P(C' | B), where C' is the event of not being awarded a credit card.

To calculate P(C' | B), we can use the complement rule and Bayes' theorem:

P(C' | B) = 1 - P(C | B) = 1 - P(C and B) / P(B)
P(C and B) = P(C | B) * P(B) = P(C | B) * 0.2

Since we are not given the value of P(C | B), we need to find it.

Now, let's consider the information that states "The company awards credit cards to 70% of those who apply." This means that P(C) = 0.7 (probability of being awarded a credit card).

The law of total probability states that:
P(C) = P(C and A) + P(C and B)

We want to find P(C and B), so we can rewrite the equation as:
P(C and B) = P(C) - P(C and A)

Using Bayes' theorem, P(C and A) = P(C | A) * P(A) = 0.95 * 0.8

Substituting the values back into the equation, we have:
P(C and B) = P(C) - P(C and A) = 0.7 - (0.95 * 0.8)

Now, substituting this value back into the equation for P(C' | B), we get:
P(C' | B) = 1 - P(C | B) = 1 - (P(C and B) / P(B)) = 1 - ((0.7 - (0.95 * 0.8)) / 0.2)

Calculating this expression will give us the probability that an applicant who is a bad credit risk will be denied a credit card.