Question

Research shows that 30% of all women taking a certain medication do not respond favorably. Use the binomial probabilities table to find the probabilities that among nine randomly selected women taking the medication

(a) at least four will respond favorably;

(b) at least four will not respond favorably.

Answer 1

Solution

Given that ,

All women taking a certain medication do not respond favorably is 0.30.

All women taking a certain medication do respond favorably is 0.70.

a)

p = 0.70

1 - p = 0.30

n = 9

Using binomial probability formula ,

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

P(X 4) = 1 - P(X < 4)

= 1 - P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

P(X = ) = 1 - ( ((9! / 0! (9-0)!) * 0.70⁰ * (0.30)^7-0 +  ((9! / 1! (9-1)!) * 0.70¹ * (0.30)^7-1 + ((9! / 2! (9-2)!) * 0.70²* (0.30)^7-2 + ((9! / 3! (9-3)!) * 0.70³ * (0.30)^7-3​​​​​​​ )

= 1 - ( 0 + 0.0004 + 0.0039 + 0.021)

Probability = 0.9747

b)

p = 0.30

1 - p = 0.70

n = 9

Using binomial probability formula ,

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

P(X 4) = 1 - P(X < 4)

= 1 - P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

P(X = ) = 1 - ( ((9! / 0! (9-0)!) * 0.30⁰ * (0.70)^7-0 +  ((9! / 1! (9-1)!) * 0.30¹ * (0.70)^7-1​​​​​​​ + ((9! / 2! (9-2)!) * 0.30²* (0.70)^7-2 + ((9! / 3! (9-3)!) * 0.30³ * (0.70)^7-3​​​​​​​ )

= 1 - (0.0404 + 0.1556 + 0.2668 + 0.2668)

Probability = 0.2703