Question

1. alison knows baseball statistics and follows major league baseball. prior to each game, she predicts which team will win and keeps track of her success rate. her overall success rate historically is 90%. let ( g = ) the number of games she gets incorrect in the next 20 games. which value of ( g ) is least likely?

a) 20
b) 19
c) 10
d) 2
e) 1

Explanation:

Step1: Define binomial parameters

Alison's incorrect prediction rate is $p = 1 - 0.90 = 0.10$, number of trials $n=20$, $G\sim Binomial(n=20,p=0.10)$.

Step2: Recall binomial probability formula

The probability mass function is $P(G=k)=\binom{n}{k}p^k(1-p)^{n-k}$, where $\binom{n}{k}=\frac{n!}{k!(n-k)!}$.

Step3: Analyze likelihood trend

For a binomial distribution with small $p$, probabilities decrease as $k$ moves far above the mean $\mu=np=20\times0.10=2$. Larger $k$ values are less likely.

Step4: Compare options

Among the choices, $G=20$ is the farthest from the mean $\mu=2$, so it has the lowest probability.

Answer:

A) 20