Chapter 6.3, Problem 86E

a.

To determine

To explain: The random variable X being binomial random variable.

Explanation of Solution

Given:

Probability of success $\left(p\right)=\frac{1}{6}$

Number of events ( n ) = 7

The random variable X being binomial random variable:

1. There are two successes and failures.
2. Candies are independent of each other.
3. Number of friends are fixed.
4. Probability of success is fixed.

Since, all the required condition for a binomial distribution are fixed. Thus, the variable X is a binomial random variable.

b.

To determine

To find: The mean and standard deviation

Answer to Problem 86E

The mean and standard deviation are 1.167 and 0.9860.

Explanation of Solution

Calculation:

The mean and standard deviation can be calculated as:

  $\begin{array}{c}\mu =np\\ =7\left(\frac{1}{6}\right)\\ =1.167\end{array}$

  $\begin{array}{c}\sigma =\sqrt{np\left(1-p\right)}\\ =\sqrt{7\left(\frac{1}{6}\right)\left(1-\frac{1}{6}\right)}\\ =0.9860\end{array}$

The mean and standard deviation are 1.167 and 0.9860.

c.

To determine

To find: Whether the friends got lucky by computing probability $P\left(X\ge 3\right)$

Answer to Problem 86E

The probability is 0.0958.

Explanation of Solution

Calculation:

  $P\left(X\ge 3\right)$ can be calculated as:

  $\begin{array}{c}P\left(X\ge 3\right)=1-P\left(X<3\right)\\ =1{-}^7{C}_r\times {\left(\frac{1}{6}\right)}^r\times {\left(1-\frac{1}{6}\right)}^{7-r}\\ =0.0958\end{array}$

The required probability is 0.0958.