Understanding the Conditional Probability Problem
We need to find the conditional probability. The condition is that the sum of the numbers appearing on two dice casts is 10. We want to find the probability that the number 5 appeared at least once, given this condition.
Determining the Conditional Sample Space
Let A be the event that the sum of the numbers is 10. The possible outcomes $(x, y)$ where $x$ is the result of the first cast and $y$ is the result of the second cast are:
- (4, 6)
- (5, 5)
- (6, 4)
The total number of outcomes where the sum is 10 is $n(A) = 3$. This is our reduced sample space.
Identifying Favorable Outcomes
Let B be the event that the number 5 appears at least once.
We need to find the outcomes in event A (sum is 10) where the number 5 appears at least once. Looking at the outcomes in A:
- (4, 6) - Does not contain 5.
- (5, 5) - Contains 5 at least once.
- (6, 4) - Does not contain 5.
The only outcome where the sum is 10 AND the number 5 appears at least once is (5, 5). So, the number of favorable outcomes is $n(A \cap B) = 1$.
Calculating the Conditional Probability
The conditional probability $P(B|A)$ is calculated as:
$ P(B|A) = \frac{n(A \cap B)}{n(A)} $Substituting the values we found:
$ P(B|A) = \frac{1}{3} $Therefore, the probability that the number 5 has appeared at least once, given that the sum is 10, is $\frac{1}{3}$.
Similar Questions
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Important Questions from Probability
Three dice are thrown. What is the probability of getting a sum which is a perfect square?
Two distinct natural numbers from 1 to 9 are picked at random. What is the probability that their product has 1 in its unit place?
Two dice are thrown. What is the probability that difference of numbers on them is 2 or 3 ?
Suppose that there is a chance for a newly constructed building to collapse, whether the design is faulty or not. The chance that the design is faulty is 10%. The chance that the building collapses is 95% if the design is faulty, otherwise it is 45%. If it is seen that the building has collapsed, then what is the probability that it is due to faulty design?
What is the probability that all three boys sit together?
