Chapter 9, Problem 15PS

a.

To determine

Find the odds against choosing a blue marble.

Answer to Problem 15PS

  $30$

Explanation of Solution

Calculation:

Consider the bag in which six marbles are red

The odds against choosing the red marbles are $4$ to $1$

Note that the odds in favor of an event occuring is the ratio of the probability that the event will occur to the probability of the event that it will not occur.

The reciprocal of this ration gives the odds against the event occuring.

Hence, if there is 1 red marble then there are 4 marbles of color other than red.

Hence, the total marbles in the bag will be

  $\frac{\left(4+1\right)}{1}\left(6\right)=5\times 6$

  $=30$

Hence, the odds against choosing a blue marble are $30$ .

b.

To determine

Find the number of marbles in the bag.

Answer to Problem 15PS

  $\frac{3}{7},\frac{7}{3}$

Explanation of Solution

Calculation:

Consider the bag containing three blue marbles and seven yellow marbles.

The probability that the marble chosen is blue is given by

  $\frac{3}{3+7}=\frac{3}{10}$

Also, the probability that the marble chosen is not blue

  $\frac{7}{3+7}=\frac{7}{10}$

Hence, the odds in favour of choosing a blue marble is

  $\frac{\frac{3}{10}}{\frac{7}{10}}=\frac{3}{7}$

Hence, the odds in favour of choosing a blue marble is $\frac{3}{7}$

Note that the odds against choosing a blue marble is given by the reciprocal of the odds in favour

Hence, the odds against choosing a blue marble is $\frac{7}{3}$

c.

To determine

Write a formula for converting the odds in favor of an event to the probability of the event.

Answer to Problem 15PS

  $P(E)=\frac{a}{a+b}$

Explanation of Solution

Calculation:

Suppose, the odds in favour of an event $E$ are

  $\frac{a}{b}$

Hence, the probability of the event $E$ is

  $P(E)=\frac{a}{a+b}$

d.

To determine

Write a formula for converting the probability of an event to the odds in favor of the event.

Answer to Problem 15PS

  $P(E)=\frac{P(E)}{1-P(E)}$

Explanation of Solution

Calculation:

Suppose, the probability of an event $E$ be $P(E)$

Hence, the odds in favour of the event is $P(E)=\frac{P(E)}{1-P(E)}$