Question

Consider a binomial probability distribution with p=0.65 and n=6. Determine the probabilities below. Round to four decimal places as needeed.

a) P(x=2)

b) P(x< or equal to1)

c) P(x>4)

Answer #1

Solution

Given that ,

p = 0.65

1 - p = 1 - 0.65 = 0.35

n = 6

Using binomial probability formula ,

P(X = x) = ((n! / x! (n - x)!) * p^{x} * (1 - p)^{n -
x}

(a)

x = 2

P(X = 2) = ((6! / 2! (6 - 2)!) * 0.65^{2} * (0.35)^{6
- 2}

= ((6! / 2! (4)!) * 0.65^{2} *
(0.35)^{4}

P(X = 2) = 0.0951

Probability = 0.0951

(b)

P(x 1) = P(x = 0) + P(x = 1)

= ((6! / 0! (6)!) * 0.65^{0} *
(0.35)^{6} + ((6! / 1! (5)!) * 0.65^{1}
* (0.35)^{5}

= 0.0223

P(x 1) = 0.0223

(c)

P(x > 4) = 1 - P(x 4)

= 1 - [P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4)]

= 1 - [((6! / 0! (6)!) * 0.65^{0} * (0.35)^{6}
+ ((6! / 1! (5)!) * 0.65^{1} *
(0.35)^{5} + ((6! / 2! (4)!) * 0.65^{2} *
(0.35)^{4}

+ ((6! / 3! (3)!) * 0.65^{3} * (0.35)^{3} + ((6!
/ 4! (2)!) * 0.65^{4} * (0.35)^{2} ]

= 1 - 0.6809

= 0.3191

P(x > 4) = 0.3191

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