JEE Main 2022 — Probability Question with Solution

Coin is tossed 5 times, so n = 5

Let, p = probability of getting heads

q = probability of getting tails.

$$\therefore$$ p + q = 1 ...... (1)

$$\therefore$$ Probability of getting 4 heads

= ⁵C₄ . p⁴ . q

And probability of getting 5 heads

= ⁵C₅ . p⁵

Given, ⁵C₄ . p⁴ . q = ⁵C₅ . p⁵

$$\Rightarrow$$ 5q = p ....... (2)

From equation (1) and (2), we get,

5q + q = 1

$$\Rightarrow$$ 6q = 1

$$\Rightarrow$$ q = $$\frac{1}{6}$$

$$\therefore$$ p = 1 $$-$$ $$\frac{1}{6}$$ = $$\frac{5}{6}$$

Now, probability of getting atmost two heads

= p (x = 0) + p (x = 1) + p (x = 2)

p (x = 0) = Getting zero head in 5 trials

= ⁵C₀ . p⁰ . q⁵

p (x = 1) = Getting one head in 5 trials

= ⁵C₁ . p¹ . q⁴

p (x = 2) = Getting two heads in 5 trials

= ⁵C₂ . p² . q³

= ⁵C₀ . q⁵ + ⁵C₁ . pq⁴ + ⁵C₂ . p²q³

$$={\left(\frac{1}{6}\right)}^5+5.\frac{5}{6}.{\left(\frac{1}{6}\right)}^4+10.{\left(\frac{5}{6}\right)}^2.{\left(\frac{1}{6}\right)}^3$$

$$=\frac{1+25+250}{6^5}=\frac{276}{6^5}$$ = $$\frac{46}{6^4}$$