Total Probability Theorem

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JEE Advanced 1994

LEVELJEE Main

An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the number obtained by adding the numbers on the two faces is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered 2, 3, 4, ..., 12 is picked and the number on the card is noted. What is the probability that the noted number is either 7 or 8?

JEE Advanced 1984

LEVELJEE Main

In a certain city only two newspapers $A$ and $B$ are published, it is known that $25%$ of the city population reads $A$ and $20%$ reads $B$ while $8%$ reads both $A$ and $B$ . It is also known that $30%$ of those who read $A$ but not $B$ look into advertisements and $40%$ of those who read $B$ but not $A$ look into advertisements while $50%$ of those who read both $A$ and $B$ look into advertisements. What is the percentage of the population that reads an advertisement?

JEE Advanced 2004

LEVELJEE Main

A box contains 12 red and 6 white balls. Balls are drawn from the box one at a time without replacement. If in 6 draws there are at least 4 white balls, find the probability that exactly one white is drawn in the next two draws. (binomial coefficients can be left as such)

JEE Advanced 2001

LEVELJEE Main

An urn contains $m$ white and $n$ black balls. A ball is drawn at random and is put back into the urn along with $k$ additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. What is the probability that the ball drawn now is white?

JEE Advanced 1988

LEVELJEE Main

Urn $A$ contains 6 red and 4 black balls and urn $B$ contains 4 red and 6 black balls. One ball is drawn at random from urn $A$ and placed in urn $B$ . Then one ball is drawn at random from urn $B$ and placed in urn $A$ . If one ball is now drawn at random from urn $A$ , the probability that it is found to be red is .........

JEE Advanced 1986

LEVELJEE Advanced

A lot contains 20 articles. The probability that the lot contains exactly 2 defective articles is 0.4 and the probability that the lot contains exactly 3 defective articles is 0.6. Articles are drawn from the lot at random one by one without replacement and are tested till all defective articles are found. What is the probability that the testing procedure ends at the twelfth testing?

JEE Advanced 2000

LEVELJEE Advanced

A coin has probability $p$ of showing head when tossed. It is tossed $n$ times. Let $p_{n}$ denote the probability that no two (or more) consecutive heads occur. Prove that $p_{1} = 1, p_{2} = 1 - p^{2}$ and $p_{n} = (1 - p) p_{n - 1} + p (1 - p) p_{n - 2}$ for all $n \geq 3$ . Prove by induction on $n$ , that $p_{n} = A α^{n} + B β^{n}$ for all $n \geq 1$ , where $α$ and $β$ are the roots of quadratic equation $x^{2} - (1 - p) x - p (1 - p) = 0$ and $A = \frac{p ^{2} + β - 1}{α β - α ^{2}}, B = \frac{p ^{2} + α - 1}{α β - β ^{2}}$ .

JEE Advanced 2000

LEVELJEE Main

A coin has probability $p$ of showing head when tossed. It is tossed $n$ times. Let $p_{n}$ denote the probability that no two (or more) consecutive heads occur. Prove that $p_{1} = 1, p_{2} = 1 - p^{2}$ and $p_{n} = (1 - p) p_{n - 1} + p (1 - p) p_{n - 2}$ for all $n \geq 3$ .

JEE Main 2018 (Paper 1)

LEVELJEE Main

A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is :

JEE Main 2019 (10 January Shift 1)

LEVELJEE Main

An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered 1,2,3,...,9 is randomly picked and the number on the card is noted. The probability that the noted number is either 7 or 8 is :

(A)

\frac{13}{36}

(B)

\frac{19}{36}

(C)

\frac{19}{72}

(D)

\frac{15}{72}

JEE Main 2019 (9 January)

LEVELJEE Main

An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red, is :

JEE Main 2019 (9 April)

LEVELJEE Main

Two newspapers A and B are published in a city. It is known that 25% of the city populations reads A and 20% reads B while 8% reads both A and B. Further, 30% of those who read A but not B look into advertisements and 40% of those who read B but not A also look into advertisements, while 50% of those who read both A and B look into advertisements. Then the percentage of the population who look into advertisement is :-

JEE Main 2023 (15 April Shift 1)

LEVELJEE Main

A bag contains 6 white and 4 black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The probability that all the balls drawn are white is

(A)

\frac{1}{4}

(B)

\frac{11}{50}

(C)

\frac{1}{5}

(D)

\frac{9}{50}

JEE Main 2025 (January)

LEVELJEE Main

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$ where $g cd (m, n) = 1$ , then $m + n$ is equal to:

JEE Main 2025 (January)

LEVELJEE Main

Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains n white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is $29/45$ then n is equal to:

JEE Main 2026 (23 January Shift 2)

LEVELJEE Main

Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls. One ball is randomly picked up from the bag B and mixed up with the balls in the bag A. Then a ball is randomly drawn from the bag A. If the probability, that the ball drawn is white, is $\frac{p}{q}$ , $gcd (p, q) = 1$ , then $p + q$ is equal to :