Understanding Conditional Probability
This problem involves conditional probability. We need to find the probability of an event (getting at least one 5) given that another event has already occurred (the sum of the numbers is 10).
Identifying Possible Outcomes (Sum = 10)
First, let's list all possible pairs of outcomes when rolling a dice twice that sum up to 10. Let the outcome be represented as (roll1, roll2).
- (4, 6)
- (5, 5)
- (6, 4)
There are 3 possible outcomes where the sum is 10. This is our reduced sample space.
Identifying Favorable Outcomes (At Least One 5)
Next, we identify which of these outcomes include the number 5 appearing at least once.
- (4, 6) - Does not contain a 5.
- (5, 5) - Contains at least one 5 (it contains two 5s).
- (6, 4) - Does not contain a 5.
Only 1 outcome, (5, 5), satisfies the condition of having at least one 5.
Calculating the Conditional Probability
The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes in the reduced sample space.
Let A be the event that the number 5 appears at least once.
Let B be the event that the sum of the numbers is 10.
We are looking for P(A | B).
Number of outcomes where sum is 10 = 3.
Number of outcomes where sum is 10 AND at least one 5 appears = 1 (which is the outcome (5, 5)).
The conditional probability is calculated as: $ P(\text{At least one 5} | \text{Sum is 10}) = \frac{\text{Number of outcomes with sum 10 and at least one 5}}{\text{Number of outcomes with sum 10}} $
$ P(A | B) = \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?
