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A bag contains 7 blue balls and 5 yellow balls. If two balls are selected at random, one by one without replacement. What is the probability that none of the selected ball is yellow?
The probability that the first ball is not yellow is $$\frac{7}{12}$$
The probability that the second ball is not yellow is $$\frac{6}{11}$$
Hence, the total probability is $$\frac{7}{12}*\frac{6}{11}=\frac{7}{22}$$
2 dice are cast. What is the probability that both the dice show the same number?
Number of favourable cases = 6
Total number of cases = 6*6 = 36
So, probability = 1/6
In how many different ways can the letters of the word 'STRESS' be arranged
Total number of letters = 6 of which 3 are the same.
Therefore, number of ways of arranging = $$\frac{6!}{3!}$$ = $$\frac{720}{6}$$ = 120
In how many ways can the letters of the word MIRACLE be arranged such that all the vowels remain together.
MIRACLE contains I,A,E as vowels apart from M, R, C, L as consonants. Group I, A, E as one unit.
Total number of ways = 5!
Now I, A, E can be arranged in 3! ways.
Total number of ways = 5!*3! = 120*6 = 720 ways
Prakhar has six biased coins with him; whenever he tosses any four coins, the probability of him getting at least one head and at least one tail is equal to 1. How many coins with Prakhar have only either heads or tails on both sides?
Let's say Prakhar has only one coin with heads on both sides. If there is only one such coin, then there would be cases in which he chooses 4 out of those six coins, but he does not pick that coin
Then, his probability of getting at least one head can not be 1.
For this probability to be 1, he must have at least three coins with heads on both sides. So, no matter what four coins he picks, he picks one coin with heads on both sides, resulting in his probability of getting at least one head 1.
The same logic applies to tails; he must have three coils with tails on both sides.
Therefore, all six coins with prakhar have either heads or tails on both sides.
Hence, Option E is the correct answer.
The probability that it will rain on Monday is 0.3. The probability that Ramya brings an umbrella on monday is 0.6.
What is the probability that Ramya does not bring an umbrella, given that it rains on Monday?
The question uses the Bayes theorem., but it can also be solved by thinking logically
Essentially, we are given that it rains, and when it rains, Ramya brings the umbrella with a probability of 0.6. Hence the probability that she does not bring the umbrella on Monday is 0.4
These are two independent events and hence do not affect each other.
The question states that is rains; the probability of Ramy not having an umbrella is independent of this and can directly be answered to be 0.4
Using Bayes theorem too, we get the same answer:$$\frac{0.4\times\ 0.6}{0.4\times\ 0.6\ +\ 0.6\times\ 0.6}=0.4$$
Hence, Option A is the correct answer.
Find the number of ways of distributing five identical chocolates and six identical cookies among three children so that each child gets at least one cookie.
The number of ways of distributing $$n$$ items among $$r$$ persons is $$^{n+r-1}C_{r-1}$$
If at least one item is given to each person, then the number of ways is $$^{n-1}C_{r-1}$$
As there are five chocolates and three children, the number of ways is $$^{n+r-1}C_{r-1}=^{5+3-1}C_{3-1}=^7C_2=21$$
For cookies, there are six cookies and we need to give at least 1 to each child, the number of ways is $$^{n-1}C_{r-1}=^{6-1}C_{3-1}=^5C_2=10$$
So, the total number of ways is $$21\times10=210$$
Hence, the answer is 210.
The faces of a numbered die are to be painted by 6 distinct colours: Red, blue, green, black, indigo, and violet. In how many ways can the faces be painted such that the red, green and blue faces share a corner of the cube?
There are eight corners in the die; we can choose any of them in 8 ways.
Say we choose the corner with numbers (3,5,1); now we have to paint them in colours red, blue and green.
This again can be done in 3! ways.
The remaining three faces are to be painted with the remaining three colours, which again can be done in three ways!
Giving us a total of $$8\times\ 6\times\ 6=288$$
Therefore, Option E is the correct answer.
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