The given probability distribution is a binomial distribution with having a ticket is the favorable incident and let's assume its probability is P.
The probability density function of a binomial distribution can be given as,
`f(k;n,p) ^nC_k P^k(1-P)^(n-k)`
In this case n = 10 and k = 4.
`0.2508 = ^10C_4 xx P^4 xx (1-P)^(10-4)`
`0.2508 = 210xx P^4xx(1-P)^6`
`P^4xx(1-P)^6 = 0.2508/210`
`(P^2(1-P)^3)^2 = 0.2508/210`
`P^2(1-P)^3 = 0.034558439`
`P^2(1-3P+3P^2-P^3) = 0.034558439`
`P^2-3P^3+3P^4-P^5 = 0.034558439`
`P^5-3P^4+3P^3-P^2+0.034558439 = 0`
You will have to solve this to find the answer for P.
The solutions for the above equation are,
1.1753 + 0.2296i
1.1753 - 0.2296i
But from above solutions only two are acceptable.
`P = 0.4021 or 0.3979`
If we take the mean value of this probability,
`P = (0.4021+0.3979) /2 = 0.4`
If we substitute 0.4 in the above equation, it would satisfy it.
Therefore, the proabability that a random man is having a red ticket is 0.4.
As a percentage the value would be 40%.