Chapter PIV, Problem 42RE

(a)

To determine

To find: the probability that the TV is won by the 1st person in line.

Answer to Problem 42RE

0.01

Explanation of Solution

Given:

The total number of individuals who join the charity is $n=100$ .

The number of balls in the box that are white is 99.

One is the number of red balls.

Formula used:

  $\text{Probability =}\frac{\text{Number of favorable outcomes for an event}}{\text{Total number of outcomes}}$

Calculation:

Winning the TV is getting the red ball in the draw.

The goal is to classify the likelihood that the TV will be won by a person on the line.

Probability is characterised as follows:

  $\text{Probability =}\frac{\text{Number of favorable outcomes for an event}}{\text{Total number of outcomes}}$

Thus,

  $\begin{array}{c}\text{Probability =}\frac{\text{Getting}\text{Red}\text{ball}}{\text{Total}\text{number}\text{of}\text{balls}}\\ \text{=}\frac{1}{100}\\ =0.01\end{array}$

Therefore, the probability that the TV is won by a person in the line is 0.01

(b)

To determine

To find: if the 3^rd person is in the line, the probability that the TV will win.

Answer to Problem 42RE

0.0098

Explanation of Solution

The goal is to find the probability that the TV will be won by the third person in the line.

The first two individuals do not get the red ball to determine the above likelihood. This implies that the probability of not receiving a red ball is $\left(1-0.01\right)=0.99$

The probability, therefore, that the third person wins the TV in the line is

  $0.99\times 0.99\times 0.01=0.0098$

Therefore, the probability that the TV is 0.0098 for the third person in the line.

(c)

To determine

To find: the probability that since no one wins, the charity decides to sell the TV.

Answer to Problem 42RE

0.3660

Explanation of Solution

The goal is to find the chance that the TV will not be won by someone on the side.

The probability that no one will get a red ball is 0.99.

Therefore, the probability of 100 people not having a red ball is ${\left(0.99\right)}^{100}=0.3660$

Therefore, the probability of no one winning the TV in the line is 0.3660.

(d)

To determine

To explain: that implies getting in line to pick a spot. Where would like to be to improve chances of winning.

Explanation of Solution

It should place in the first position to increase the probability of winning when the chosen balls are replaced. This is because the probability of winning is the same in every place. The likelihood of winning in the first trial is equal to the likelihood of winning in the last trial. Therefore, the best location to increase the probability of winning is first place.

(e)

To determine

To explain: the position in line will select in order to maximize the possibility.

Explanation of Solution

It does not matter where stand in the line, to maximise the chance of winning if the picked balls are not replaced. This implies that position doesn't matter. It is because there are random draws.