Advertisements
Advertisements
Question
A pair of dice is thrown. If sum of the numbers is an even number, what is the probability that it is a perfect square?
Advertisements
Solution
When two dice are thrown simultaneously, the sample space is
S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
∴ n(S) = 36
Let A be the event that sum of the numbers is an even number.
∴ A = {(1, 1), (1, 3), (1, 5) (2, 2), (2, 4), (2, 6), (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (4, 6), (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6)}
∴ n(A) = 18
∴ P(A) = `("n"("A"))/("n"("S")) = 18/36`
Let B be the event that sum of outcomes is a perfect square
B = {(1, 3), (2, 2), (3, 1), (3, 6), (4, 5), (5, 4), (6, 3)}
∴ n(B) = 7
∴ P(B) = `("n"("B"))/("n"("S")) = 7/36`
∴ A ∩ B is the event that the sum of the numbers is an even perfect square.
∴ A ∩ B = {(1, 3), (2, 2), (3, 1)}
∴ n(A ∩ B) = 3
∴ P(A ∩ B) = `("n"("A" ∩ "B"))/("n"("S")) = 3/36`
Now, Probability that the sum of the numbers is a Perfect square given that it is even is given by,
`"P"("B"/"A") = ("P"("A" ∩ "B"))/("P"("A")`
= `(3/36)/(18/36)`
= `3/18`
= `1/6`
APPEARS IN
RELATED QUESTIONS
Evaluate P(A ∪ B), if 2P(A) = P(B) = `5/13` and P(A | B) = `2/5`
A black and a red dice are rolled.
Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.
A fair die is rolled. Consider events E = {1, 3, 5}, F = {2, 3} and G = {2, 3, 4, 5} Find P ((E ∪ F)|G) and P ((E ∩ G)|G)
A box has 20 pens of which 2 are defective. Calculate the probability that out of 5 pens drawn one by one with replacement, at most 2 are defective.
In a college, 70% of students pass in Physics, 75% pass in Mathematics and 10% of students fail in both. One student is chosen at random. What is the probability that:
(i) He passes in Physics and Mathematics?
(ii) He passes in Mathematics given that he passes in Physics.
(iii) He passes in Physics given that he passes in Mathematics.
Two dice are thrown simultaneously, If at least one of the dice show a number 5, what is the probability that sum of the numbers on two dice is 9?
An urn contains 4 black, 5 white, and 6 red balls. Two balls are drawn one after the other without replacement, What is the probability that at least one ball is black?
From a pack of well-shuffled cards, two cards are drawn at random. Find the probability that both the cards are diamonds when the first card drawn is replaced in the pack
A year is selected at random. What is the probability that it is a leap year which contains 53 Sundays
Choose the correct alternative:
If A and B are any two events, then the probability that exactly one of them occur is
Choose the correct alternative:
If two events A and B are independent such that P(A) = 0.35 and P(A ∪ B) = 0.6, then P(B) is
In a multiple-choice question, there are three options out of which only one is correct. A person is guessing the answer at random. If there are 7 such questions, then the probability that he will get exactly 4 correct answers is ______
If X denotes the number of ones in five consecutive throws of a dice, then P(X = 4) is ______
Let A and B be two events. If P(A) = 0.2, P(B) = 0.4, P(A ∪ B) = 0.6, then P(A|B) is equal to ______.
If P(A) = `1/2`, P(B) = 0, then `P(A/B)` is
For a biased dice, the probability for the different faces to turn up are
| Face | 1 | 2 | 3 | 4 | 5 | 6 |
| P | 0.10 | 0.32 | 0.21 | 0.15 | 0.05 | 0.17 |
The dice is tossed and it is told that either the face 1 or face 2 has shown up, then the probability that it is face 1, is ______.
It is given that the events A and B are such that P(A) = `1/4, P(A/B) = 1/2` and `P(B/A) = 2/3`, then P(B) is equal to ______.
Read the following passage:
|
Recent studies suggest the roughly 12% of the world population is left-handed.
Assuming that P(A) = P(B) = P(C) = P(D) = `1/4` and L denotes the event that child is left-handed. |
Based on the above information, answer the following questions:
- Find `P(L/C)` (1)
- Find `P(overlineL/A)` (1)
- (a) Find `P(A/L)` (2)
OR
(b) Find the probability that a randomly selected child is left-handed given that exactly one of the parents is left-handed. (2)
Three friends go to a restaurant to have pizza. They decide who will pay for the pizza by tossing a coin. It is decided that each one of them will toss a coin and if one person gets a different result (heads or tails) than the other two, that person would pay. If all three get the same result (all heads or all tails), they will toss again until they get a different result.
- What is the probability that all three friends will get the same result (all heads or all tails) in one round of tossing?
- What is the probability that they will get a different result in one round of tossing?
- What is the probability that they will need exactly four rounds of tossing to determine who would pay?
