Question

Suppose a loaded die has the following probability model:

If this die is thrown and the top face shows an odd number, what is the probability that the die shows a 1?

a. $0.10$

b. $0.17$

c. $0.30$

d. $0.50$

e. $0.60$

Short answer

The probability that the die shows a$1$is$0.60$

Step-by-step solution

Given information

Probability model for loaded die:

Calculation

The addition rule for disjoint or mutually exclusive events is as follows:

$P(\mathrm{A}\cup \mathrm{B})=P(\mathrm{A}\text{or}\mathrm{B})=P(\mathrm{A})+P(\mathrm{B})$

Definition of conditional probability:

$P(\mathrm{B}\mid \mathrm{A})=\frac{P(\mathrm{A}\cap \mathrm{B})}{P(\mathrm{A})}=\frac{P(\mathrm{A}\text{and}\mathrm{B})}{P(\mathrm{A})}$

From the table,

Probability to roll a 1

$P\left(1\right)=0.3$

When it comes to the chances of rolling an odd number$(1,3or5)$

For disjoint events, use the following addition rule:

$P\left(\text{odd}\right)=P\left(1\right)+P\left(3\right)+P\left(5\right)=0.3+0.1+0.1=0.5$

Definition of conditional probability:

$P(1\mid \text{odd})=\frac{P\left(\text{l andodd}\right)}{P\left(\text{odd}\right)}=\frac{0.3}{0.5}=\frac{3}{5}=0.60$

Therefore, the conditional probability to roll 1 is$0.60$