Question

A fair coin is tossed 20 times. The probability that ‘head’ will appear exactly 4 times in the first ten tosses, and ‘tail’ will appear exactly 4 times in the next ten tosses is ______ (round off to 3 decimal places).

Answer 1

Concept:

Binomial probability refers to probability of exactly ‘r’ success in ‘n’ number of trials of a experiment which has two possible outcomes.

If the probability of success is ‘p’ in individual trial and the probability of failure is ‘q’ then

 P(r) = ⁿC_r p^rq^n-

Where, n = no. of trials, p = probability of success of random variable, q = probability of failure of random variable, r = number of occurrence of random variable

Calculation:

Probability of 4 heads in first 10 tosses and 4 tails in next ten tosses will be given by

P (4 head in 10 tosses) × P (4 tail in 10 tosses)

Probability of 4 heads in first 10 tosses

n = 10, \({\rm{p\;}} = {\rm{\;}}\frac{1}{2},\;q = 1 - \frac{1}{2} = \frac{1}{2}\), r = 4

P (4 head in 10 tosses) = ¹⁰C₄ × 0.5⁴ × 0.5⁶ = 0.205

Probability of 4 tails in last 10 tosses

n = 10, \({\rm{p\;}} = {\rm{\;}}\frac{1}{2},\;q = 1 - \frac{1}{2} = \frac{1}{2}\), r = 4

P (4 tail in 10 tosses) = ¹⁰C₄ × 0.5⁴ × 0.5⁶ = 0.205

∴ Probability of 4 heads in first 10 tosses and 4 tails in next ten tosses = 0.205 × 0.205 = 0.0420