Chapter 10.6, Problem 21E

(a)

To determine

To explain:

The statement:

‘The first four kids were all the boys, so the next will probably be a girl’.

Answer to Problem 21E

The statement states that after having four children would be not a girl. The fifth child would be either a girl or a boy.

Hence, the statement is wrong.

Explanation of Solution

Given information:

The probability of each one: $p=0.5$

In this situation, there are chances of having a female child and a male child. It’s asked to check out the validity of the statement.

Having a male child is independent of having a female child.

The statement states that after having four children would be not a girl. The fifth child would be either a girl or a boy.

Hence, the statement is wrong.

(b)

To determine

To calculate:

The probability of having four male children and then female child

Answer to Problem 21E

The probability of having four male children and then female child is $0.03125$ .

Explanation of Solution

Given information:

  $\begin{array}{l}n=5\\ p=0\\ k=5\end{array}$

Formula used:

  $P\left(ksuccesses\right)=C{}_k{p}^k{\left(1-p\right)}^{n-k}$

Where,

  $P$: Probability of Success

Calculation:

For binomial experiment, the probability of exactly $n$ trials and $k$ successes is:

  $P\left(ksuccesses\right)=C{}_k{p}^k{\left(1-p\right)}^{n-k}$

Put the values in the formula:

  $\begin{array}{l}P\left(k=5\right)=C{}_5{\left(0.5\right)}^5{\left(0.5\right)}^0\\ P\left(k=5\right)=0.03125\end{array}$

The answer is $P\left(k=5\right)=0.03125$ .

(c)

To determine

To graph:

The histogram of probability distribution

Answer to Problem 21E

The histogram of the probability distribution:

  

Graph: Histogram

Explanation of Solution

Given information:

  $x$ : Number of children

Interpretation:

  $P\left(x\right)=C{}_k{p}^k{\left(1-p\right)}^{n-k}$

Put the values in the formula:

  $\begin{array}{l}P\left(x\right)={0.5}^x\times 0.5\\ P\left(x\right)={0.5}^{\left(x+1\right)}\end{array}$

The histogram of the probability distribution:

  

Graph: Histogram

The histogram of the probability distribution is shown as above.