Chapter 10, Problem 6RE

Solution

To determine: The probability that a player wins the price.

The probability that a player wins the price.is 0.042.

Given information:

The bag contains 10 marbles out of which 5 are red, 3 are white, and 2 are blue.

Formula used:

  $\mathrm{P}=\frac{E}{S}$   ...... (1)

Here, E is the favourable outcomes and S is the total possible outcomes.

And,

Use general multiplication rule:

P (A and B) = P(A)×P(B|A) = P(B)×P(A|B)    ...... (2)

Calculation:

The probability of selecting first a red marble, then a white marble, and then a blue marble.

Let

R =Red of first draw

W =White on second draw

B =Blue on third draw

5 of the 10 marbles in the bag are red.

Substitute 5 for $E$ and 10 for $S$ in equation (1)

  $\mathrm{P}(R)=\frac{5}{10}$

When the first selected marbles were red, then 3 of the 9 remaining marbles in the bag are white.

Substitute 3 for $E$ and 9 for $S$ in equation (1)

  $\mathrm{P}(W|R)=\frac{3}{9}$

When the first selected marbles was red and the second selected marbles was white , then 2 of the 8 remaining marbles are blue.

Substitute 2 for $E$ and 8 for $S$ in equation (1)

  $\mathrm{P}(B|RW)=\frac{2}{8}$

Substitute $\frac{5}{10}$ for $\mathrm{P}(R)$ , $\frac{3}{9}$ for $\mathrm{P}(W|R)$ and $\frac{2}{8}$ for $\mathrm{P}(B|RW)$ in equation (2)

  $\begin{array}{c}P\left(RWB\right)=\frac{5}{10}\times \frac{3}{9}\times \frac{2}{8}\\ =\frac{30}{720}\\ =\frac{1}{24}\\ =0.042\end{array}$

Hence, the probability that a player wins the price.is 0.042.