A balanced coin is tossed 8 times. Let X denotes the number of heads. Calculate the following: / ‘n Gebalanseerde muntstuk word 8 keer gewerp. Laat X die aantal munte aandui. Bereken die volgende:
1.1 P(X = 5)
1.2 P(X 2)
1.3 P(X > 2)
To calculate these probabilities, we can use the binomial probability formula:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
where n is the number of trials (8 in this case), k is the number of successful outcomes (heads in this case), and p is the probability of success on a single trial (0.5 for a balanced coin).
1.1 P(X=5)
P(X=5) = (8 choose 5) * 0.5^5 * 0.5^3
P(X=5) = 56 * 0.03125 * 0.125
P(X=5) = 0.21875
1.2 P(X < 2)
P(X=0) = (8 choose 0) * 0.5^0 * 0.5^8
P(X=0) =1 * 1 * 0.00390625 = 0.00390625
P(X=1) = (8 choose 1) * 0.5^1 * 0.5^7
P(X=1) = 8 * 0.5 * 0.0078125 = 0.03125
P(X<2) = P(X=0) + P(X=1) = 0.00390625 + 0.03125 = 0.03515625
1.3 P(X > 2)
P(X>2) = 1 - P(X<2)
P(X>2) = 1 - 0.03515625
P(X>2) = 0.96484375
Therefore:
1.1 P(X=5) = 0.21875
1.2 P(X < 2) = 0.03515625
1.3 P(X > 2) = 0.96484375