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1

WB JEE 2022

MCQ (Single Correct Answer)
English
Bengali

A, B, C are mutually exclusive events such that $$P(A) = {{3x + 1} \over 3}$$, $$P(B) = {{1 - x} \over 4}$$ and $$P(C) = {{1 - 2x} \over 2}$$. Then the set of possible values of x are in

A
[0, 1]
B
$$\left[ {{1 \over 3},{1 \over 2}} \right]$$
C
$$\left[ {{1 \over 3},{2 \over 3}} \right]$$
D
$$\left[ {{1 \over 3},{{13} \over 3}} \right]$$

A, B ও C এমন তিনটি পরস্পর বিচ্ছিন্ন ঘটনা যে $$P(A) = {{3x + 1} \over 3}$$, $$P(B) = {{1 - x} \over 4}$$ এবং $$P(C) = {{1 - 2x} \over 2}$$ হয়। সেক্ষেত্রে x-এর সম্ভাব্য মানের সেট হবে

A
[0, 1]
B
$$\left[ {{1 \over 3},{1 \over 2}} \right]$$
C
$$\left[ {{1 \over 3},{2 \over 3}} \right]$$
D
$$\left[ {{1 \over 3},{{13} \over 3}} \right]$$
2

WB JEE 2021

MCQ (Single Correct Answer)
English
Bengali
Four persons A, B, C and D throw and unbiased die, turn by turn, in succession till one gets an even number and win the game. What is the probability that A wins the game if A begins?
A
$${1 \over 4}$$
B
$${1 \over 2}$$
C
$${7 \over 15}$$
D
$${8 \over 15}$$

Explanation

Even numbers on die are 2, 4, 6 and odd numbers on die are 1, 3, 5.

$$\therefore$$ P(even) = $${3 \over 6} = {1 \over 2}$$ and P(odd) = $${3 \over 6} = {1 \over 2}$$

According to the question,

$$P({A_{(wins)}}) = {1 \over 2} + {\left( {{1 \over 2}} \right)^4} \times {1 \over 2} + {\left( {{1 \over 2}} \right)^8} \times {1 \over 2} + {\left( {{1 \over 2}} \right)^{12}} \times {1 \over 2} + ...\infty $$

$$ = {1 \over 2} + {\left( {{1 \over 2}} \right)^5} + {\left( {{1 \over 2}} \right)^9} + {\left( {{1 \over 2}} \right)^3} + ...\infty $$

$$ = {{{1 \over 2}} \over {1 - {{\left( {{1 \over 2}} \right)}^4}}} = {{{1 \over 2}} \over {{{15} \over {16}}}} = {1 \over 2} \times {{16} \over {16}} = {8 \over {15}}$$
A, B, C ও D চারজন ব্যাক্তি একটি নিরপেক্ষ ছক্কা পরপর ছুঁড়তে থাকলাে যতক্ষণ না একজন জোড়সংখ্যা ছুঁড়লাে এবং জিতলাে। যদি A ছোড়া শুরু করে তবে তার জেতার সম্ভাবনা হবে ?
A
$${1 \over 4}$$
B
$${1 \over 2}$$
C
$${7 \over 15}$$
D
$${8 \over 15}$$

Explanation

ডাই এর জোড় সংখ্যা হল 2, 4, 6 এবং ডাই এর উপর বিজোড় সংখ্যা হল 1, 3, 5।

$$\therefore$$ P(জোড়) = $${3 \over 6} = {1 \over 2}$$ এবং P(বিজোড়) = $${3 \over 6} = {1 \over 2}$$

প্রশ্ন অনুযায়ী,

$$P({A_{(\text{জেতার})}}) = {1 \over 2} + {\left( {{1 \over 2}} \right)^4} \times {1 \over 2} + {\left( {{1 \over 2}} \right)^8} \times {1 \over 2} + {\left( {{1 \over 2}} \right)^{12}} \times {1 \over 2} + ...\infty $$

$$ = {1 \over 2} + {\left( {{1 \over 2}} \right)^5} + {\left( {{1 \over 2}} \right)^9} + {\left( {{1 \over 2}} \right)^3} + ...\infty $$

$$ = {{{1 \over 2}} \over {1 - {{\left( {{1 \over 2}} \right)}^4}}} = {{{1 \over 2}} \over {{{15} \over {16}}}} = {1 \over 2} \times {{16} \over {16}} = {8 \over {15}}$$
3

WB JEE 2020

MCQ (Single Correct Answer)
English
Bengali
A rifleman is firing at a distant target and has only 10% chance of hitting it. The least number of rounds he must fire to have more than 50% chance of hitting it at least once, is
A
5
B
7
C
9
D
11

Explanation

Given, $$P = {{10} \over {100}} = {1 \over {10}}$$

$$q = 1 - P = 1 - {1 \over {10}} = {9 \over {10}}$$

$$P(X \ge 1) = 1 - P(X = 0)$$

$$P(X \ge 1) = 1 - {}^n{C_0}{\left( {{1 \over {10}}} \right)^0}{\left( {{9 \over {10}}} \right)^n}$$

$$1 - {\left( {{9 \over {10}}} \right)^n} \ge {1 \over 2}$$

$${\left( {{9 \over {10}}} \right)^n} \le {1 \over 2}$$

For n = 7 the least number of round to must so fire.

এক বন্দুক চালকের লক্ষ্যে আঘাত করার সম্ভাবনা 10%। লক্ষ্যে অন্তত একবার আঘাত করার সম্ভাবনা 50% এর বেশি হওয়ার জন্য কমপক্ষে যতবার গুলি ছুঁড়তে হবে তার সংখ্যা হল -

A
5
B
7
C
9
D
11

Explanation

এখানে $$P = {{10} \over {100}} = {1 \over {10}},\,q = 1 - p = {9 \over {10}}$$

$$50\% = {1 \over 2}$$

$$\therefore$$ $$1 - {q^n} > {1 \over 2} \Rightarrow {q^n} < {1 \over 2}$$

বা, $${\left( {{9 \over {10}}} \right)^n} < {1 \over 2}$$

ইহা সম্ভব যখন n-এর মান কমপক্ষে 7 হয়।

4

WB JEE 2020

MCQ (Single Correct Answer)
English
Bengali
Four persons A, B, C and D throw an unbiased die, turn by turn, in succession till one gets an even number and win the game. What is the probability that A wins if A begins?
A
$${1 \over 4}$$
B
$${1 \over 2}$$
C
$${7 \over 12}$$
D
$${8 \over 15}$$

Explanation

Four person A, B, C, D throw an unbiased die, turn by turn, in succession till one gets an even number and win the game.

Probability of an even number = $${1 \over 2}$$

$$ \therefore $$ $$P(A) = P(B) = P(C) = P(D) = {1 \over 2}$$

$$P(\overline A ) = P(\overline B ) = P(\overline C ) = P(\overline D ) = {1 \over 2}$$

Required probability

$$ = P(A) + P(\overline A )P(\overline B )P(\overline C )P(\overline D )P(A) + P(\overline A )P(\overline B )P(\overline C )P(\overline D )P(\overline A )P(\overline B )P(\overline C )P(\overline D )P(A) + ...$$

$$ = {1 \over 2} + {\left( {{1 \over 2}} \right)^5} + {\left( {{1 \over 2}} \right)^9} + ...$$

$$ = {{1/2} \over {1 - {{(1/2)}^4}}} = {{1/2} \over {15/16}} = {8 \over {15}}$$

A, B, C ও D চারজন একটি পক্ষপাতহীন ছক্কা পরপর নিক্ষেপ করবে এবং যে প্রথম জোড় সংখ্যা নিক্ষেপ করবে সে জিতবে, যদি A প্রথম নিক্ষেপ করে তবে তার জেতার সম্ভাবনা হবে

A
$${1 \over 4}$$
B
$${1 \over 2}$$
C
$${7 \over {12}}$$
D
$${8 \over {15}}$$

Explanation

জোড় পড়ার সম্ভাবনা $${3 \over 6} = {1 \over 2}$$

এবং বিজোড় পড়ার সম্ভাবনা $$ = {1 \over 2}$$

এক্ষেত্রে সম্ভাবনা

$$ = P(A) + P(\overline A \cap \overline B \cap \overline C \cap \overline D \cap A) + P(\overline A \cap \overline B \cap \overline C \cap \overline D \cap \overline A \cap \overline B \cap \overline C \cap \overline D \cap A) + .....\infty $$

$$ = {1 \over 2} + {\left( {{1 \over 2}} \right)^5} + {\left( {{1 \over 2}} \right)^9} + .....\infty $$

$$ = {{{1 \over 2}} \over {1{{\left( {{1 \over 2}} \right)}^4}}} = {{{2^3}} \over {{2^4} - 1}} = {8 \over {15}}$$

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