Eight coins are tossed 25,600 times. The average number of eight heads is

Concept:

Binomial Distribution

If p is the probability that an event will happen to say in a single trial (called the probability of success), then q = 1 – p, is the probability that event will fail to happen (called probability of failure). If random variable X represents the number of success that occurs in trials. Then according to Binomial theorem.

\(P\left( {X = r} \right) = {n_{{C_r}}} × {\left( p \right)^r} × {\left( q \right)^{n - r}}\), r = 0, 1, 2, 3……………..

Mean = n × p, Variance = n × p × q

Calculation:

Given: 

Total number of times coins are tossed, N = 25600

Number of coins, n = 8

\(Probability\left( {success} \right) = p\left( H \right) = \frac{1}{2} = q\)

\(Probability\left( {failure} \right) = p\left( T \right) = \frac{1}{2} = q\)

Probability of getting 8 heads when 8 coins are tossed, \(P\left( {X = 8} \right) = {8_{{C_8}}} \times {\left( {\frac{1}{2}} \right)^8} \times {\left( {\frac{1}{2}} \right)^0} = \frac{1}{{256}}\)

Expected frequency or average number of eight head = N × P (X = 8) = \(25600 \times \frac{1}{{256}} = 100\)