Question

4747​% of men consider themselves professional baseball fans. you randomly select 10 men and ask each if he considers himself a professional baseball fan. find the probability that the number who consider themselves baseball fans is​ (a) exactly​ five, (b) at least​ six, and​ (c) less than four.

Answer 1

The situation represents a binomial probabilty with the probability of success (p) = 47% or 0.47 and the number of trials (n) = 10.

The probability of a binomial distribution is given by:


`P(X)=\left(^n_x)p^x(1-p)^(n-x)`


Part A:

The probability that the number who consider themselves baseball fans is​ exactly​ five is given by:


`P(5)=\left(^(10)_(\,5)\right)(0.47)^5(1-0.47)^(10-5) \\ \\ =252(0.022935)(0.041820)=\bold{0.2417}`



Part B:

The probability that the number who consider themselves baseball fans is​ at least​ six is given by:


`P(X\geq6)=P(6)+P(7)+P(8)+P(9)+p(10) \\ \\ =\left(^(10)_(\,6)\right)(0.47)^6(1-0.47)^(10-6)+\left(^(10)_(\,7)\right)(0.47)^7(1-0.47)^(10-7) \\ +\left(^(10)_(\,8)\right)(0.47)^8(1-0.47)^(10-8)+\left(^(10)_(\,9)\right)(0.47)^9(1-0.47)^(10-9) \\ +\left(^(10)_(10)\right)(0.47)^(10)(1-0.47)^(10-10) \\ \\ =210(0.010779)(0.078905)+120(0.005066)(0.148877) \\ +45(0.002381)(0.2809)+10(0.001119)(0.53)+1(0.000526)(1) \\ \\ =0.1786+0.0905+0.0301+0.0059+0.0005=\bold{0.3056}`

This can be approximated using normal distribution as I will illustrate in part c.



Part C:

The probability that the number who consider themselves baseball fans is​ less than four is given by:


`P(X\ \textless \ 4)=P(0)+P(1)+P(2)+P(3)`

We can approximate this using normal distribution by subtracting 0.5 from the least value and adding 0.5 to the greatest value.

This gives
`P(0-0.5\ \textless \ X\ \textless \ 3+0.5)=P(-0.5\ \textless \ X\ \textless \ 3.5)`

The mean of a binomial distribution is given by
`\mu=np`, while the standard deviation is given by
`\sigma= √(np(1-p))`.

Thus,


`\mu=10(0.47)=4.7 \ and \ \sigma=√(10(0.47)(0.53))=1.578`

The probability of a nomal distribution between two values (a, b) is given by:


`P(a\ \textless \ X\ \textless \ b)=P\left( (b-\mu)/(\sigma) \right)-P\left( (a-\mu)/(\sigma) \right)`

Thus,


`P(-0.5\ \textless \ X\ \textless \ 3.5)=P\left( (3.5-4.7)/(1.578) \right)-P\left( (-0.5-4.7)/(1.578) \right) \\ \\ =P(-0.7605)-P(-3.295)=0.2235-0.0005=\bold{0.223}`

Answer 2

Given:
p = 47% = 0.47, the probability that a man considers himself a professional baseball fan.
q = 1 - p = 0.53, the probability that a man does not consider himself a professional baseball fan.
n = 10, the number of men surveyed.

(a) Calculate the probability that exactly 5 of 10 men surveyed consider themselves as professional baseball fans.
P(5 of 10) = ₁₀C₅ p⁵q⁵ = 252*0.47⁵*0.53⁵ = 0.242

Answer: 0.242 or 24.2%

(b) Calculate the probability of at least 6 out of 10.
P(at least 6 of 10)
= ₁₀C₆ p⁶q⁴ + ₁₀C₇ p⁷q³ + ₁₀C₈ p⁸q² + ₁₀C₉ p⁹q + ₁₀C₁₀ p¹⁰ q⁰
= 0.1786 + 0.0905 + 0.0301 + 0.0059 + 0.0
= 0.3057

Answer: 0.3057 or 30.6%

(c) Calculate the probability of less than 4 of 10.
P(less than 4 of 10)
= ₁₀C₁ pq⁹ + ₁₀C₂ p²q⁸ + ₁₀C₃ p³q⁷
= 0.0155 + 0.0619 + 0.1464
= 0.2238

Answer: 0.2238 or 22.4%