Answer: 5 times
Step-by-step explanation: The expected value of a random variable is the average value that it would take over many repeated trials. In this case, the random variable is the number of times the spinner lands on A in 20 spins. To calculate the expected value, we need to know the probability of each possible outcome and multiply it by the corresponding value. For example, the probability of landing on A zero times is (3/4)^20, which is about 0.003%, and the value is 0. So we multiply 0.003% by 0 and get 0. Similarly, the probability of landing on A once is 20 x (1/4) x (3/4)^19, which is about 0.016%, and the value is 1. So we multiply 0.016% by 1 and get 0.016%. We do this for all possible outcomes from 0 to 20, and add them up to get the expected value. The formula is:
E(X) = sum of P(X=x) x x for x = 0 to 20
where X is the number of times the spinner lands on A, P(X=x) is the probability of landing on A x times, and x is the possible value.
When we simplify this formula, we get:
E(X) = (1/4) x sum of x for x = 0 to 20
which is equivalent to:
E(X) = (1/4) x (0 + 1 + 2 + … + 20)
Using the formula for the sum of an arithmetic series, we get:
E(X) = (1/4) x (20/2) x (0 + 20)
which simplifies to:
E(X) = (1/4) x 10 x 20
which equals:
E(X) = 5
This means that the expected number of times the spinner lands on A in 20 spins is 5.
Hope this helps, and have a great day! =)