Bernoulli's Theorem.

Bernoulli's Theorem :-

The probability of x successes in n trials is equal to ⁿC_x p^x q^{n - x}  wherxe p is the probability of success and q is the probability of failure. 

Proof : Since p is the probability of success, the probability of x success is p•p•p••• x times = p^x 

Also x success imply (n-x) failures and since q is the probability of failure, the probability of (n-x) failures is q^{n - x} . 

By the multiplication rule, the probability of the Simultaneous happening is p^x q ^{n - x} . But x successes in n trials can occure in   ⁿCx way and all these cases are favorable to the event. 

Hence by the addition rule, the probability of x successes out of n trials is given by

P^x q^{n - x} + p^xq ^{n - x}. . . + ⁿC_x times = ⁿC_x p^x q^{n - x}

This proves Bernoulli's theorem. 

An illustration of the theorem

Suppose a coin is tossed 3 times and we wish to know the probability of getting 2 heads. 

The possibilities are as follows:

   HHH, HHT,  HTH, HTT,  THH,  THT,  TTH,  TTT

We see that there are eight possibilities, out of which only 3 possibilities are favourable to the event. They are HHT,  HTH,   THH. 

Thus the probability of getting 2 heads is 3/8.

Now we shall obtain the probability of the event by Bernoulli's theorem. 

Number of trails n=3.

Number of success (p) = probability of getting a head =1/2

Probability of failure (q) = 1-p=1/2

Therefore 

ⁿC_x p^x q^{n - x} = ³C₂ (1/2)² (1/2)¹ = 3*1/4*1/2=3/8