With n = 20 and p = 0.60, estimate P(fewer than 8).

Question

help me please

Answer 1

This is a binomial distribution, I guess, so we have (p+(1-p))ⁿ=1.

For the given parameters we have (0.60+0.40)²⁰ as the distribution formula we need P(X<8).

The terms in the expansion are p²⁰+20p¹⁹(1-p)+(20*19/2)p¹⁸(1-p)²+...+(1-p)²⁰.

X is the exponent of the p term, so in words the expansion means P(X=20)+P(X=19)+...+P(X=0).

For P(X<8) we need the sum of P(X=0) up to P(X=7). The coefficients are ⁿCᵣ values where n=20 and 0≤r≤7 as follows:

1, 20, 190, 1140, 4845, 15504, 38760, 77520.

The p and 1-p terms are respectively:

1.099512E-8, 1.649267E-8, 2.473901E-8, 3.710852E-8, 5.566278E-8, 8.349416E-8, 12.524125E-8, 18.786187E-8.

Put the terms with their coefficients and we get 0.021029 so P(X<8)=2.1% approx.