Chapter 3, Problem 3.1P

Two fair dice are rolled. What is the conditional probability that at least one lands on 6 given that the dice land on different numbers?

To determine

To Calculate: The conditional probability that atleast one lands on 6 given that the dice land on different numbers.

Answer to Problem 3.1P

The Conditional probability that atleast one dice lands on 6 given that the dice land on different numbers is $\frac{1}{6}$.

Explanation of Solution

Given information:

Tossing of two dice and number on both the dice are different.

Concept and Formula Used:

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

Conditional Probability- Probability of an event when one event already happened.

$\text{P(E/F)}=\frac{\text{P(E}\cap \text{F)}}{\text{P(F)}}$

Calculation:

The tossing of two dice result in 36 outcomes.

Let ‘E’ be the event that atleast one dice lands on 6.

Sample space for event ‘E’ are $(1,6),(2,6),(3,6),(4,6),(5,6)\&(6,6)$

Let ‘F’ be the event that both the numbers are different on the dice.

Sample space for event ‘F’ are

$\begin{array}{l}(1,2),(1,3),(1,4),(1,5),(1,6)\\ (2,1),(2,3),(2,4),(2,5),(2,6)\\ (3,1),(3,2),(3,4),(3,5),(3,6)\\ (4,1),(4,2),(4,3),(4,5),(4,6)\\ (5,1),(5,2),(5,3),(5,4),(5,6)\\ (6,1),(6,2),(6,3),(6,4),(6,5)\end{array}$

We have to find the probability of getting atleast one dice lands on 6 and given that numbers on both the dice are different.

$\text{P(E/F)}=\text{P(Event E when event F is given)}=\frac{\text{P(E}\cap \text{F)}}{\text{P(F)}}$.

$\text{E}\cap \text{F}=\text{common of event E and F}=(1,6),(2,6),(3,6),(4,6)\&(5,6)$ . So,

$\text{P(E}\cap \text{F)}=\frac{5}{36},\text{P(F)=}\frac{30}{36}$.

Therefore,

$\begin{array}{c}\text{P(E/F)}=\frac{\text{P(E}\cap \text{F)}}{\text{P(F)}}\\ =\frac{\frac{5}{36}}{\frac{30}{36}}\\ =\frac{5}{36}\times \frac{36}{30}\\ =\frac{1}{6}\end{array}$

Conclusion:

The Conditional probability that atleast one dice lands on 6 given that the dice land on different numbers is $\frac{1}{6}$.