In a binomial distribution,n=8 and π=.35. Find the probabilities of the following events. (Round your answers to 4 decimal places.)
(a)x=1
(b)x≤4
(c)x≥5
To find the probabilities of the events in a binomial distribution, we can use the formula:
P(x) = (nCx)(π^x)(1-π)^(n-x)
(a) To find P(x=1):
P(x=1) = (8C1)(0.35^1)(1-0.35)^(8-1)
= (8)(0.35)(0.65)^7
≈ 0.2258
(b) To find P(x≤4):
P(x≤4) = P(x=0) + P(x=1) + P(x=2) + P(x=3) + P(x=4)
= (8C0)(0.35^0)(1-0.35)^(8-0) + (8C1)(0.35^1)(1-0.35)^(8-1) + (8C2)(0.35^2)(1-0.35)^(8-2) + (8C3)(0.35^3)(1-0.35)^(8-3) + (8C4)(0.35^4)(1-0.35)^(8-4)
= (1)(1)(0.65)^8 + (8)(0.35)(0.65)^7 + (28)(0.35^2)(0.65)^6 + (56)(0.35^3)(0.65)^5 + (70)(0.35^4)(0.65)^4
≈ 0.6554
(c) To find P(x≥5):
P(x≥5) = P(x=5) + P(x=6) + P(x=7) + P(x=8)
= (8C5)(0.35^5)(1-0.35)^(8-5) + (8C6)(0.35^6)(1-0.35)^(8-6) + (8C7)(0.35^7)(1-0.35)^(8-7) + (8C8)(0.35^8)(1-0.35)^(8-8)
= (56)(0.35^5)(0.65)^3 + (28)(0.35^6)(0.65)^2 + (8)(0.35^7)(0.65)^1 + (1)(0.35^8)(0.65)^0
≈ 0.9798
Thus, the probabilities are approximately:
(a) P(x=1) ≈ 0.2258
(b) P(x≤4) ≈ 0.6554
(c) P(x≥5) ≈ 0.9798
To find the probabilities of the events in a binomial distribution, we can use the formula:
P(x) = (nCx)(π^x)((1-π)^(n-x))
where n is the number of trials, π is the probability of success, x is the number of successes, and nCx is the binomial coefficient, which can be calculated using the formula:
nCx = n! / (x! * (n-x)!)
Now let's calculate the probabilities for the given events:
(a) P(x=1):
To find the probability that x=1, we can substitute n=8, π=0.35, and x=1 into the probability formula:
P(x=1) = (8C1)(0.35^1)((1-0.35)^(8-1))
Using the binomial coefficient formula, we have:
8C1 = 8! / (1! * (8-1)!) = 8
Substituting the values:
P(x=1) = 8(0.35^1)(0.65^7) ≈ 0.3087
Therefore, the probability P(x=1) is approximately 0.3087.
(b) P(x≤4):
To find the probability that x≤4, we need to sum the probabilities of x=0, 1, 2, 3, and 4. We can use the probability formula to calculate each individual probability:
P(x=0) = (8C0)(0.35^0)(0.65^8)
P(x=1) = (8C1)(0.35^1)(0.65^7)
P(x=2) = (8C2)(0.35^2)(0.65^6)
P(x=3) = (8C3)(0.35^3)(0.65^5)
P(x=4) = (8C4)(0.35^4)(0.65^4)
Using the binomial coefficient formula, we can calculate each nCx value:
8C0 = 1
8C1 = 8
8C2 = 8! / (2! * (8-2)!) = 28
8C3 = 8! / (3! * (8-3)!) = 56
8C4 = 8! / (4! * (8-4)!) = 70
Substituting the values:
P(x≤4) = P(x=0) + P(x=1) + P(x=2) + P(x=3) + P(x=4)
= (1)(0.35^0)(0.65^8) + (8)(0.35^1)(0.65^7) + (28)(0.35^2)(0.65^6) + (56)(0.35^3)(0.65^5) + (70)(0.35^4)(0.65^4)
≈ 0.4237
Therefore, the probability P(x≤4) is approximately 0.4237.
(c) P(x≥5):
To find the probability that x≥5, we need to sum the probabilities of x=5, 6, 7, and 8. We can use the probability formula to calculate each individual probability, similar to part (b).
Using the same binomial coefficient formulas:
8C5 = 8! / (5! * (8-5)!) = 56
8C6 = 8! / (6! * (8-6)!) = 28
8C7 = 8! / (7! * (8-7)!) = 8
8C8 = 1
Substituting the values:
P(x≥5) = P(x=5) + P(x=6) + P(x=7) + P(x=8)
= (56)(0.35^5)(0.65^3) + (28)(0.35^6)(0.65^2) + (8)(0.35^7)(0.65^1) + (1)(0.35^8)
≈ 0.1173
Therefore, the probability P(x≥5) is approximately 0.1173.