JEE Advanced Probability Questions with Solutions & PYQ Practice

Q 1 :

A die is thrown. Let $A$ be the event that the number obtained is greater than 3. Let $B$ be the event that the number obtained is less than 5. Then $P (A \cup B)$ is [2008]

$\frac{3}{5}$
$0$
$1$
$\frac{2}{5}$

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4

(3)

$A (number is greater than 3) = {4, 5, 6}$

$\Rightarrow P (A) = \frac{3}{6} = \frac{1}{2}$

$B (number is less than 5) = {1, 2, 3, 4} \Rightarrow P (B) = \frac{4}{6} = \frac{2}{3}$

$∴$ $A \cap B = {4} \Rightarrow P (A \cap B) = \frac{1}{6}$

$∴$ $P (A \cup B) = P (A) + P (B) - P (A \cap B)$

$= \frac{1}{2} + \frac{2}{3} - \frac{1}{6} = \frac{3 + 4 - 1}{6} = 1$

Q 2 :

If $P (B) = \frac{3}{4}, P (A \cap B \cap \bar{C}) = \frac{1}{3}$ and $P (\bar{A} \cap B \cap \bar{C}) = \frac{1}{3},$ then $P (B \cap C)$ is [2003]

$\frac{1}{12}$
$\frac{1}{6}$
$\frac{1}{15}$
$\frac{1}{9}$

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4

(1)

$Given : P (B) = \frac{3}{4}, P (A \cap B \cap \bar{C}) = \frac{1}{3}$

$P (\bar{A} \cap B \cap \bar{C}) = \frac{1}{3}$

[IMAGE 372]

$From above venn diagram, we see$

$B \cap C = B - (A \cap B \cap \bar{C}) - (\bar{A} \cap B \cap \bar{C})$

$\Rightarrow P (B \cap C) = P (B) - P (A \cap B \cap \bar{C}) - P (\bar{A} \cap B \cap \bar{C})$

$\Rightarrow P (B \cap C) = \frac{3}{4} - \frac{1}{3} - \frac{1}{3}$ $= \frac{9 - 4 - 4}{12} = \frac{1}{12}$

Q 3 :

A number is chosen at random from the set {1, 2, 3, ..., 2000}. Let $p$ be the probability that the chosen number is a multiple of 3 or a multiple of 7. Then the value of 500 $p$ is ________. [2021]

(214)

$E = set of numbers divisible by 3$

$E = {3, 6, 9, 12, \dots, 1998}$

$∴$ $n (E) = 666$

$F = set of numbers divisible by 7$

$F = {7, 14, 21, \dots, 1995}$

$∴$ $n (F) = 285$

$E \cap F = set of numbers divisible by 3 and 7$ $= {21, 42, \dots, 1995}$

$∴$ $n (E \cup F) = 666 + 285 - 95 = 856$

$Required probability = \frac{856}{2000} = P$

$So, 500 P = \frac{856}{2000} \times 500 = 214$

Q 4 :

Let $E, F$ and $G$ be three events having probabilities $P (E) = \frac{1}{8}, P (F) = \frac{1}{6}, P (G) = \frac{1}{4},$ and let $P (E \cap F \cap G) = \frac{1}{10} .$

For any event $H$ , if $H^{c}$ denotes its complement, then which of the following statements is(are) TRUE? [2021]

$P (E \cap F \cap G^{c}) \leq \frac{1}{40}$
$P (E^{c} \cap F \cap G) \leq \frac{1}{15}$
$P (E \cup F \cup G) \leq \frac{13}{24}$
$P (E^{c} \cap F^{c} \cap G^{c}) \leq \frac{5}{12}$

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Select one or more options

(1, 2, 3)

$Given that$

$P (E) = \frac{1}{8}, P (F) = \frac{1}{6}, P (G) = \frac{1}{4}, P (E \cap F \cap G) = \frac{1}{10}$

(3) $P (E \cup F \cup G)$ $= P (E) + P (F) + P (G)$ $- P (E \cap F) - P (F \cap G) - P (G \cap E) + P (E \cap F \cap G)$

$= \frac{1}{8} + \frac{1}{6} + \frac{1}{4} - \sum P (E \cap F) + \frac{1}{10}$

$= \frac{13}{24} + \frac{1}{10} - \sum P (E \cap F)$

$\Rightarrow P (E \cup F \cup G) \leq \frac{13}{24}$ $[Option (3) is correct]$

(4) $Now, P (E^{c} \cap F^{c} \cap G^{c})$

$= 1 - P (E \cup F \cup G) \geq 1 - \frac{13}{24}$

$\Rightarrow P (E^{c} \cap F^{c} \cap G^{c}) \geq \frac{11}{24}$ $[Option (4) is incorrect]$

(1) $∵$ $P (E) \geq P (E \cap F \cap G) + P (E \cap F \cap G^{c})$

$\Rightarrow \frac{1}{8} \geq P (E \cap F \cap G^{c}) + \frac{1}{10}$

$\Rightarrow \frac{1}{8} - \frac{1}{10} \geq P (E \cap F \cap G^{c})$

$\Rightarrow \frac{1}{40} \geq P (E \cap F \cap G^{c})$ $[Option (1) is correct]$

(2) $∵$ $P (F) \geq P (E^{c} \cap F \cap G) + P (E \cap F \cap G)$

$\Rightarrow \frac{1}{6} \geq P (E^{c} \cap F \cap G) + \frac{1}{10}$

$\Rightarrow \frac{1}{6} - \frac{1}{10} \geq P (E^{c} \cap F \cap G)$

$\Rightarrow \frac{4}{60} \geq P (E^{c} \cap F \cap G)$

$\Rightarrow \frac{1}{15} \geq P (E^{c} \cap F \cap G)$ $[Option (2) is correct]$

Q 5 :

Two players, $P_{1}$ and $P_{2}$ , play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let $x$ and $y$ denote the readings on the die rolled by $P_{1}$ and $P_{2}$ , respectively. If $x > y$ , then $P_{1}$ scores 5 points and $P_{2}$ scores 0 point. If $x = y$ , then each player scores 2 points. If $x < y$ , then $P_{1}$ scores 0 point and $P_{2}$ scores 5 points. Let $X_{i}$ and $Y_{i}$ be the total scores of $P_{1}$ and $P_{2}$ , respectively, after playing the $i^{th}$ round. [2022]

List - I List - II

(I) Probability of $(X_{2} \geq Y_{2})$ is (P) $\frac{3}{8}$

(II) Probability of $(X_{2} > Y_{2})$ is (Q) $\frac{11}{16}$

(III) Probability of $(X_{3} = Y_{3})$ is (R) $\frac{5}{16}$

(IV) Probability of $(X_{3} > Y_{3})$ (S) $\frac{355}{864}$

(T) $\frac{77}{432}$

The correct option is:

(I) → (Q); (II) → (R); (III) → (T); (IV) → (S)
(I) → (Q); (II) → (R); (III) → (T); (IV) → (T)
(I) → (P); (II) → (R); (III) → (Q); (IV) → (S)
(I) → (P); (II) → (R); (III) → (Q); (IV) → (T)

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(1)

$P (X_{i} > Y_{i}) + P (X_{i} < Y_{i}) + P (X_{i} = Y_{i}) = 1$

$and P (X_{i} > Y_{i}) = P (X_{i} < Y_{i}) = p$

$For i = 2$

$P (X_{2} = Y_{2}) = P (5, 5) + P (4, 4)$

$= \frac{5}{12} \times \frac{5}{12} \times 2 + \frac{1}{6} \times \frac{1}{6}$ $= \frac{25}{72} + \frac{1}{36} = \frac{27}{72} = \frac{3}{8}$

$P (X_{2} > Y_{2}) = P (10, 0) = \frac{5}{12} \times \frac{5}{12} + \frac{5}{12} \times \frac{1}{6} \times 2 = \frac{5}{16}$

$P (X_{2} \geq Y_{2}) = \frac{5}{16} + \frac{3}{8} = \frac{11}{16}$

$I \to Q, I I \to R$

$For i = 3$

$P (X_{3} = Y_{3}) = P (6, 6) + P (7, 7)$

$= \frac{1}{6 \times 6} + \frac{5}{12} \times \frac{1}{6} \times \frac{5}{12} \times 6 = \frac{77}{432}$

$P (X_{3} > Y_{3}) = \frac{1}{2} (1 - \frac{77}{432}) = \frac{355}{864}$

$I I I \to T, I V \to S$

Topic Question Set

Q 1 :

A die is thrown. Let $A$ be the event that the number obtained is greater than 3. Let $B$ be the event that the number obtained is less than 5. Then $P (A \cup B)$ is [2008]

Q 2 :

If $P (B) = \frac{3}{4}, P (A \cap B \cap \bar{C}) = \frac{1}{3}$ and $P (\bar{A} \cap B \cap \bar{C}) = \frac{1}{3},$ then $P (B \cap C)$ is [2003]

Q 3 :

A number is chosen at random from the set {1, 2, 3, ..., 2000}. Let $p$ be the probability that the chosen number is a multiple of 3 or a multiple of 7. Then the value of 500 $p$ is ________. [2021]

Q 4 :

Let $E, F$ and $G$ be three events having probabilities $P (E) = \frac{1}{8}, P (F) = \frac{1}{6}, P (G) = \frac{1}{4},$ and let $P (E \cap F \cap G) = \frac{1}{10} .$

For any event $H$ , if $H^{c}$ denotes its complement, then which of the following statements is(are) TRUE? [2021]

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	List - I		List - II
(I)	Probability of $(X_{2} \geq Y_{2})$ is	(P)	$\frac{3}{8}$
(II)	Probability of $(X_{2} > Y_{2})$ is	(Q)	$\frac{11}{16}$
(III)	Probability of $(X_{3} = Y_{3})$ is	(R)	$\frac{5}{16}$
(IV)	Probability of $(X_{3} > Y_{3})$	(S)	$\frac{355}{864}$
		(T)	$\frac{77}{432}$

Topic Question Set

Q 1 : A die is thrown. Let A be the event that the number obtained is greater than 3. Let B be the event that the number obtained is less than 5. Then P(A∪B) is [2008]

Q 2 : If P(B)=34, P(A∩B∩C¯)=13 and P(A¯∩B∩C¯)=13, then P(B∩C) is [2003]

Q 3 : A number is chosen at random from the set {1, 2, 3, ..., 2000}. Let p be the probability that the chosen number is a multiple of 3 or a multiple of 7. Then the value of 500p is ________. [2021]

Q 4 : Let E,F and G be three events having probabilities P(E)=18, P(F)=16, P(G)=14, and let P(E∩F∩G)=110. For any event H, if Hc denotes its complement, then which of the following statements is(are) TRUE? [2021]

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Q 1 :

A die is thrown. Let $A$ be the event that the number obtained is greater than 3. Let $B$ be the event that the number obtained is less than 5. Then $P (A \cup B)$ is [2008]

Q 2 :

If $P (B) = \frac{3}{4}, P (A \cap B \cap \bar{C}) = \frac{1}{3}$ and $P (\bar{A} \cap B \cap \bar{C}) = \frac{1}{3},$ then $P (B \cap C)$ is [2003]

Q 3 :

A number is chosen at random from the set {1, 2, 3, ..., 2000}. Let $p$ be the probability that the chosen number is a multiple of 3 or a multiple of 7. Then the value of 500 $p$ is ________. [2021]

Q 4 :

Let $E, F$ and $G$ be three events having probabilities $P (E) = \frac{1}{8}, P (F) = \frac{1}{6}, P (G) = \frac{1}{4},$ and let $P (E \cap F \cap G) = \frac{1}{10} .$

For any event $H$ , if $H^{c}$ denotes its complement, then which of the following statements is(are) TRUE? [2021]