Question: In a lottery game, a single ball is drawn at random from a container that contains 25 identical…

In a lottery game, a single ball is drawn at random from a container that contains 25 identical balls numbered from 1 through 25. Use the equation P(AuB) = P(A) + P(B) - P(AnB), where A and B are any events, to compute the probability that the number drawn is prime or greater than 18.

The probability that the number drawn is prime or greater than 18 is . (Type an integer or a decimal.)

Solution

To solve this problem, let’s define the events: - Let \( A \) be the event that the number drawn is prime. - Let \( B \) be the event that the number drawn is greater than 18. We need to find \( P(A \cup B) \), the probability that the number drawn is prime or greater than 18, using the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] First, determine \( P(A) \), the probability of drawing a prime number. The prime numbers between 1 and 25 are: 2, 3, 5, 7, 11, 13, 17, 19, 23. There are 9 prime numbers. \[ P(A) = \frac{9}{25} \] Next, determine \( P(B) \), the probability of drawing a number greater than 18. The numbers greater than 18 are: 19, 20, 21, 22, 23, 24, 25. There are 7 numbers greater than 18. \[ P(B) = \frac{7}{25} \] Now, determine \( P(A \cap B) \), the probability of drawing a number that is both prime and greater than 18. The prime numbers greater than 18 are: 19, 23. There are 2 such numbers. \[ P(A \cap B) = \frac{2}{25} \] Substitute these probabilities into the formula: \[ P(A \cup B) = \frac{9}{25} + \frac{7}{25} - \frac{2}{25} \] Simplify the expression: \[ P(A \cup B) = \frac{9 + 7 - 2}{25} = \frac{14}{25} \] Therefore, the probability that the number drawn is prime or greater than 18 is \(\frac{14}{25}\) or 0.56.