there is a stack of 10 cards, each given a different number from 1 to 10. suppose we select a card randomly…

there is a stack of 10 cards, each given a different number from 1 to 10. suppose we select a card randomly from the stack, replace it, and then randomly select another card. what is the probability that the first card is an even number and the second card is greater than 7? write your answer as a fraction in simplest form.

there is a stack of 10 cards, each given a different number from 1 to 10. suppose we select a card randomly from the stack, replace it, and then randomly select another card. what is the probability that the first card is an even number and the second card is greater than 7? write your answer as a fraction in simplest form.

Answer

Explanation:

Step1: Calculate probability of first - card being even

The even numbers from 1 to 10 are 2, 4, 6, 8, 10. So there are 5 even numbers out of 10. The probability of selecting an even - numbered card on the first draw, $P(A)=\frac{5}{10}=\frac{1}{2}$.

Step2: Calculate probability of second - card being greater than 7

The numbers greater than 7 from 1 to 10 are 8, 9, 10. So there are 3 numbers greater than 7 out of 10. The probability of selecting a card greater than 7 on the second draw, $P(B)=\frac{3}{10}$.

Step3: Use the multiplication rule for independent events

Since the card is replaced, the two events are independent. For independent events $A$ and $B$, $P(A\cap B)=P(A)\times P(B)$. $P(A\cap B)=\frac{1}{2}\times\frac{3}{10}=\frac{3}{20}$.

Answer:

$\frac{3}{20}$