Chapter 5.3, Problem 65E

a.

To determine

To find:the probability that the result for “a good chance” in the given table.

Answer to Problem 65E

  $P\left(\text{a good chance}|\text{Female}\right)=0.2813$ .

Explanation of Solution

Given:

A survey of 4826 randomly selected young adults as in the given table:

Opinion	Female	Male	Total
Almost no chance	93	98	194
Some chance but probably not	426	286	712
A 50-50 chance	696	720	1416
A good chance	663	758	1421
Almost certain	486	597	1083
Total	2367	2459	4826

Concept used:

Probability is the number of favourable outcomes divided by the number of possible outcomes.

Thus, Definition condition probability:

  $P\left(A|B\right)=\frac{P\left(A\text{ and }B\right)}{P\left(B\right)}$

Calculation:

According to the question, the person is female. Now to find the probability that the result for “a good chance”.

Therefore,

  $P\left(\text{a good chance and Female}\right)=\frac{\text{favourable outcomes}}{\text{possible outcomes}}=\frac{663}{4826}$

  $P\left(\text{Female}\right)=\frac{\text{favourable outcomes}}{\text{possible outcomes}}=\frac{2357}{4826}$

Thus, the conditional probability becomes:

  $\begin{array}{l}P\left(\text{a good chance}|\text{Female}\right)=\frac{P\left(\text{a good chance and Female}\right)}{P\left(\text{Male}\right)}\\ =\frac{663/4826}{2357/4826}=\frac{663}{2357}\approx 0.2813\end{array}$

Hence,the probability that the result for “a good chance” is $P\left(\text{a good chance}|\text{Female}\right)=0.2813$ .

b.

To determine

To find: the probability that the result for “a good chance” in the given table.

Answer to Problem 65E

  $P\left(\text{ a good chance}\right)=0.2944$ .

Explanation of Solution

Given:

A survey of 4826 randomly selected young adults as in the given table:

Opinion	Female	Male	Total
Almost no chance	93	98	194
Some chance but probably not	426	286	712
A 50-50 chance	696	720	1416
A good chance	663	758	1421
Almost certain	486	597	1083
Total	2367	2459	4826

Concept used:

Probability is the number of favourable outcomes divided by the number of possible outcomes.

Thus, Definition condition probability:

  $P\left(A|B\right)=\frac{P\left(A\text{ and }B\right)}{P\left(B\right)}$

Calculation:

The objective is to find the probability that the result for “a good chance”.

Therefore,

  $P\left(\text{ a good chance}\right)=\frac{\text{favourable outcomes}}{\text{possible outcomes}}=\frac{1421}{4826}\approx 0.2944$

c.

To determine

To check:whether the events “a good chance” and “female” are independent.

Answer to Problem 65E

Yes, independent.

Explanation of Solution

Given:

  $P\left(\text{a good chance}/\text{Female}\right)=\frac{663}{2357}\approx 0.2813$

  $P\left(\text{ a good chance}\right)=\frac{\text{favourable outcomes}}{\text{possible outcomes}}=\frac{1421}{4826}\approx 0.2944$ .

Calculation:

From part (a) and part (b),

  $P\left(\text{a good chance}/\text{Female}\right)=\frac{663}{2357}\approx 0.2813$

  $P\left(\text{ a good chance}\right)=\frac{\text{favourable outcomes}}{\text{possible outcomes}}=\frac{1421}{4826}\approx 0.2944$ .

There probability should be identical, thus they are independent.