00:02
In this question we have given for binomial probability distribution, p is equal to 0 .5 and n is equal to 12.
00:16
We have to find the probability that the number of success is equal to 7 and the probability that number of success is fewer than 6.
00:26
So first of all, as we know that q is equal to 1 minus p.
00:35
Therefore here we have q is equal to 1 minus 0 .5.
00:41
That is equal to 0 .5.
00:43
Now we know that the probability distribution formula is given as p of x, here take it as r, is equal to ncr multiplied with the probability.
00:57
P raised to the power r multiplied with q raised to the part n minus r so from let us name this equation as equation one therefore from equation one we have p of x is equal to r is equal to 12 c r multiplied with value of p is 0 .5 raised to the power r multiplied with value of p is 0 .5 raised to the power r multiplied with 0 .5 raised to the part 12 minus r.
01:33
Here i have substituted the non values.
01:36
Now let us consider the first case that is number of successes is equals to 7.
01:55
That is p of r is equals to 7.
02:00
So if we name this equation as equation 2 then from equation 2 we here we have 12 c 7 multiplied with 0 .5 raised to the power 7 multiplied with 0 .5 raised to the power 12 minus 7.
02:18
Now it implies probability at r is equal to 7 is equal to 12c7 multiplied with 0 .5 raised to the power 12.
02:35
Since we know that a rest to the power m multiplied with a to the power n is equal to a rest to the power m plus n.
02:47
Here i have used this property to simplify this expression.
02:52
Now on further simplifying it we can write it as probability of r is equal to 7 is equal to 12 factorial divided by 7 factorial multiple with 12 minus 7 factorial...