JEE Advanced · Mathematics · 32. Probability

The minimum number of times a fair coin needs to be tossed so that the probability of getting at least two heads is at least $0.96$ , is

A 9
B 11
C 15
D 8

Verified Solution

Answer & Solution

Correct Answer

(D) 8

Step-by-step Solution

Detailed explanation

P (2) + P (3) \dots \geq 0 .96

1 - P (0) - P (1) \geq 0 .96

1 - 0 .96 \geq^{n} C_{0} {(\frac{1}{2})}^{n} +^{n} C_{1} {(\frac{1}{2})}^{n}

\Rightarrow (0 .04) 2^{n} \geq 1 + n

\Rightarrow 4 {.2}^{n} \geq (n + 1) 100

\Rightarrow 2^{n + 2} \geq (n + 1) 100

As,

2^{10} \geq 900

\Rightarrow n = 8

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

A bag contains $N$ balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For $i=1,2,3$, let $W_i, G_i$, and $B_i$ denote the events that the ball drawn in the $i^{\text {th }}$ draw is a white ball, green ball, and blue ball, respectively. If the probability $P\left(W_1 \cap G_2 \cap B_3\right)=\frac{2}{5 N}$ and the conditional probability $P\left(B_3 \mid W_1 \cap G_2\right)=\frac{2}{9}$, then $N$ equals ________.

JEE Advanced 2024 Medium

Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

Column I	Column II
(A)The number of permutations containing the word ENDEA, is	(p) $5 !$
(B) The number of permutations in which the letter E occurs in the first and the last positions, is	(q) $2 \times 5!$
(C) The number of permutations in which none of the letters D, L, N occurs in the last five positions, is	(r) $7 \times 5!$
(D) The number of permutations in which the letters A, E, O occur only in odd positions, is	(s) $21 \times 5!$

JEE Advanced 2008 Medium

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The minimum number of times a fair coin needs to be tossed so that the probability of getting at least two heads is at least 0.96, is

Step-by-step Solution

The minimum number of times a fair coin needs to be tossed so that the probability of getting at least two heads is at least $0.96$ , is