Question

An individual can win a prize by drawing the highest numbered card out of five cards. After each draw, the card is replaced and the cards are shuffled for another chance. If the prize is not won, the card is again replaced and mixed in with the other cards. If a prize is won, the highest card is replaced and the five cards are shuffled for a new game. If an individual makes seven draws, what is the probability the individual will win a prize exactly two times?

A. 2 factorial over 7 factorial
B. 7 choose 2 times 0.2 squared times 0.8 to the fifth power
C. 0.2 squared over 0.8 to the fifth power
D. 2(0.2)(0.8)
E. (0.2)2(0.8)5

Answer 1

The correct answer is B. 7 choose 2 times 0.2 squared times 0.8 to the fifth power, as this scenario represents a binomial probability problem where the formula P(X = k) = C(n, k) * p^k * (1-p)^(n-k) is applied for n=7 draws, k=2 successes, and p=0.2 probability of winning.

Step-by-step explanation:

To calculate the probability of the individual winning a prize exactly two times in seven draws, we can use the binomial probability formula. This scenario is a classic binomial problem because each draw is independent and has two outcomes: win (with highest card) or lose.

The formula for the binomial probability is P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where:

• C(n, k) is the number of combinations of n things taken k at a time.
• p is the probability of winning (which in this case is 1/5 or 0.2 since there are 5 cards).
• n is the total number of draws.
• k is the number of successes (winning the prize).

In this case, n=7 (seven draws), k=2 (winning twice), and p=0.2 (probability of winning a single draw).

The number of ways to win twice in seven draws is given by the number of combinations, which is 7 choose 2 (7C2). The probability of winning twice is p^2, and the probability of losing the other five times is (1-p)^5.

Therefore, the correct answer is B. 7 choose 2 times 0.2 squared times 0.8 to the fifth power.

Answer 2

B

Step-by-step explanation: