This problem can be solved by the binomial probability concept. In a binomial experiment, where the probability of success in a trial is p, and the probability of failure is q, the probability of exactly x successes is given by:
P(x) = C(n, x)p^xq^(n-x)
C(n,x) is the combination of n trials selecting x at a time. It is defined as:
C(n,x) = n!/(x!(n-x)!)
Success is being a baseball fan; failure is not being a baseball fan.
p = 0.59 = the probability of being a baseball fan ---------59% = 59/100 = 0.59
q = 1-p = 1 - 0.59 = 0.41 = the probability of not being a baseball fan.
n = 10 = the total number of trials. For P(5), x = 5 = the exact number of successes
1. P(5) = C(10,5)(0.59)^5(0.41)^5 = 252(0.07149)(0.01159) = 0.2088
Remember that C(10,5) = 10!/(5!(10-5)!) = 10!/(5!5!) = (10 x 9 x 8 x 7 x 6)/(5 x 4 x 3 x 2 x 1) = 252
You can also get this figure from the calculator by pressing 10, then nCr, then 5 on your calculator.
Therefore the probability that the number who consider themselves fans are exactly 5 is 0.2088.
2. P(at least 6) = P(6) + P(7) + P(8) + P(9) + P(10)
= C(10,6)(0.59)^6(0.41)^4 + C(10,7)(0.59)^7(0.41)^3 + C(10,8)(0.59)^8(0.41)2 + C(10,9)(0.59)^9(0.41)^1 + (10,10)(0.51)^10(0.41)^0
= 0.2503 + 0.2058 + 0.1111 + 0.0355 + 0.0051
= 0.6078
Therefore, the probability that at least 6 men consider themselves as fans is 0.6078
3. P(less than 4) = P(0) + P(1) + P(2) + P(3)
= C(10,0)(0.59)^0(0.41)^10 + C(10,1)(0.59)^1(0.41)^9 + C(10,2)(0.59)^2(0.41)^8 + C(10,3)(0.59)^3(0.41)^7
= 0.0001 + 0.0019 + 0.0125 + 0.0480
= 0.0625
The probability that less than 4 people will consider themselves as fans = 0.0625.
Method 2
You can also solve this problem by using Microsoft Excel or the Binomial probabilities chart. However, it is very important that you know what you are looking for. I prefer the Microsoft Excel.
For each of the probabilities, open Excel, click on the fx, then statistical, binomial distribution. In the dialog box, enter the number of successes in the number_s box, the number of trials (in this case 10), the probability of success in the probability_s box (in this case 0.59), and TRUE or FALSE in the Cumulative box. For example, for P(5), you would enter FALSE, but for P(less than 4 ) you would enter TRUE after entering 3 in the number_s box. Using Microsoft Excel is easier, but you must know exactly what you are looking for.
