Mathematics, 08.07.2019 19:00 mikeymanfs

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.

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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

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

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

0.2238 or 22.4%

Answer from: Coltong121

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 $\sigma= \sqrt{np(1-p)}$ .

Thus,

$\mu=10(0.47)=4.7 \ and \ \sigma=\sqrt{10(0.47)(0.53)}=1.578$

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

$P(a\ \textless \ X\ \textless \ b)=P\left( \frac{b-\mu}{\sigma} \right)-P\left( \frac{a-\mu}{\sigma} \right)$

Thus,

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

Answer from: Quest

maybe showing the graph would

step-by-step explanation:

-387129 iq

Answer from: Quest

Y’all got the baby baby money in

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