Question

1. there are 10 red and 20 blue balls in a box. a ball is chosen at random and it is noted whether it is red. the process repeats, returning the ball 10 times. calculate the expected value and the standard deviation of this game.

Explanation:

Step1: Calculate the probability of choosing a red ball

The total number of balls is $10 + 20=30$. The probability $p$ of choosing a red ball is $p=\frac{10}{30}=\frac{1}{3}$. The number of trials $n = 10$.

Step2: Calculate the expected value

The formula for the expected value $E(X)$ of a binomial distribution is $E(X)=np$. Substituting $n = 10$ and $p=\frac{1}{3}$, we get $E(X)=10\times\frac{1}{3}=\frac{10}{3}\approx3.33$.

Step3: Calculate the standard deviation

The formula for the standard deviation $\sigma$ of a binomial distribution is $\sigma=\sqrt{np(1 - p)}$. Substitute $n = 10$ and $p=\frac{1}{3}$, then $1-p = 1-\frac{1}{3}=\frac{2}{3}$. So $\sigma=\sqrt{10\times\frac{1}{3}\times\frac{2}{3}}=\sqrt{\frac{20}{9}}=\frac{\sqrt{20}}{3}=\frac{2\sqrt{5}}{3}\approx1.49$.

Answer:

Expected value: $\frac{10}{3}\approx3.33$, Standard - deviation: $\frac{2\sqrt{5}}{3}\approx1.49$