Probability Refer to the results from the 150 subjects in Cumulative Review Exercise 5.

a.Find the probability that if 1 of the 150 subjects is randomly selected, the result is a woman who spent the money.

b.Find the probability that if 1 of the 150 subjects is randomly selected, the result is a woman who spent the money or was given a single 100-yuan bill.

c.If two different women are randomly selected, find the probability that they both spent the money.

A table is devised that shows the number of women who spent/kept the money depending on whether they received a single bill or smaller bills.

a.

The total number of women sampled is equal to 150.

The number of women who spent the money is equal to:

\(60 + 68 = 128\)

The probability of selecting a woman who spent the money is equal to:

\(\begin{aligned}{c}P = \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{women}}\;{\rm{who}}\;{\rm{spent}}\;{\rm{the}}\;{\rm{money}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{women}}}}\\ = \frac{{128}}{{150}}\\ = 0.853\end{aligned}\)

Thus, the probability of selecting a woman who spent the money is equal to 0.853.

b.

Let A be the event of selecting a woman who spent the money.

Let B be the event of selecting a woman who received a single 100 Yuan bill.

The probability of selecting a woman who spent the money or received a single 100 Yuan bill is denoted by:

\(P\left( {A\;or\;B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A\;and\;B} \right)\)

The probability of occurrence of event A is equal to:

\(\begin{aligned}{c}P\left( A \right) = \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{women}}\;{\rm{who}}\;{\rm{spent}}\;{\rm{the}}\;{\rm{money}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{women}}}}\\ = \frac{{128}}{{150}}\\ = 0.853\end{aligned}\)

The probability of occurrence of B is equal to:

\(\begin{aligned}{c}P\left( B \right) = \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{women}}\;{\rm{who}}\;{\rm{got}}\;{\rm{a}}\;{\rm{single}}\;{\rm{100}}\;{\rm{Yuan}}\;{\rm{bill}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{women}}}}\\ = \frac{{60 + 15}}{{150}}\\ = \frac{{75}}{{150}}\\ = 0.5\end{aligned}\)

The probability of occurrence of both A and B is equal to:

\(\begin{aligned}{c}P\left( {A\;{\rm{and}}\;B} \right) = \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{women}}\;{\rm{who}}\;{\rm{spent}}\;{\rm{the}}\;{\rm{money}}\;{\rm{and}}\;{\rm{got}}\;{\rm{a}}\;{\rm{single}}\;{\rm{100}}\;{\rm{Yuan}}\;{\rm{bill}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{women}}}}\\ = \frac{{60}}{{150}}\\ = 0.4\end{aligned}\)

Thus, the probability of occurrence of A or B is equal to:

\(\begin{aligned}{c}P\left( {A\;or\;B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A\;and\;B} \right)\\ = 0.853 + 0.5 - 0.4\\ = 0.953\end{aligned}\)

The probability of selecting a woman who spent the money and received a single 100 Yuan bill is equal to 0.953.

c.

It is given that two different women are to be randomly selected.

This implies that sampling is done without replacement.

Thus, the probability of selecting two different women who spent the money is equal to:

\(\begin{aligned}{c}P = \frac{{128}}{{150}} \times \frac{{127}}{{149}}\\ = 0.853 \times 0.852\\ = 0.727\end{aligned}\)

Therefore, the probability of selecting two different women who spent the money is equal to 0.727.