ProbabilitymediumPYQ · JEE Main · 2 Apr 2026 · Shift 2 (Afternoon)Free

Coin Probability of Exactly 30 Points: m + n = 107 | JEE 2026

JEE Maths question with a full step-by-step solution.

Question

A man throws a fair coin repeatedly. He gets 10 points for each head and 5 points for each tail. If the probability that he gets exactly 30 points is

\dfrac{m}{n}

\gcd(m,n)=1

, then

m+n

is equal to

53

55

107

correct

105

Solution

Step 1: To total exactly 30 points, the (heads, tails) combinations solving

10h+5t=30

are:

6

tails;

1

head +

4

tails;

2

heads +

2

tails;

3

heads. The probability sums the arrangements of each case:

P=\left(\tfrac12\right)^6+\frac{5!}{1!\,4!}\left(\tfrac12\right)^5+\frac{4!}{2!\,2!}\left(\tfrac12\right)^4+\left(\tfrac12\right)^3.

Step 2: Evaluate each term:

P=\frac{1}{64}+\frac{5}{32}+\frac{3}{8}+\frac{1}{8}.

Step 3: Common denominator

64

P=\frac{1+10+24+8}{64}=\frac{43}{64}=\frac{m}{n}.

(Cross-check by the recursion

q_k=\tfrac12 q_{k-1}+\tfrac12 q_{k-2}

for reaching a multiple of 5:

q_6=\tfrac{43}{64}

.) Step 4: With

m=43,\ n=64

m+n=43+64=107.

Correct answer: (3)

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