A 2005 Gallup Poll found that 7% of teenagers (ages 13 to 17) suffer from arachn

Question

A 2005 Gallup Poll found that 7% of teenagers (ages 13 to 17) suffer from arachnophobia and are extremely afraid of spiders. At a summer camp there are 10 teenagers sleeping in each tent. Assume that these 10 teenagers are independent of each other.

(a) Calculate the probability that at least one of them suffers from arachnophobia.

(b) Calculate the probability that exactly 2 of them suffer from arachnophobia.

(c) Calculate the probability that at most 1 of them suffers from arachnophobia.

(d) If the camp counselor wants to make sure no more than 1 teenager in each tent is afraid of spiders, does it seem reasonable for him to randomly assign teenagers to tents?

Explanation / Answer

A 2005 Gallup Poll found that 7% of teenagers (ages 13 to 17) suffer from arachnophobia and are extremely afraid of spiders. At a summer camp there are 10 teenagers sleeping in each tent. Assume that these 10 teenagers are independent of each other.

a). Calculate the probability that at least one of them suffers from arachnophobia.

P( atleast 1) = P( x 1)

= 1-P( x=0) = 1-0.4840

=0.5160

(b) Calculate the probability that exactly 2 of them suffer from arachnophobia.

P( x=2) =0.1234

(c) Calculate the probability that at most 1 of them suffers from arachnophobia.

P( atmost 1) = P( x 1)

= P( x=0)+P( x=1) =0.4840+0.3643 =0.8483

(d) If the camp counselor wants to make sure no more than 1 teenager in each tent is afraid of spiders, does it seem reasonable for him to randomly assign teenagers to tents?

P(no more than 1)=0.8483

The probability is large, it seems not reasonable for him to randomly assign teenagers to tents.

n=10,p=0.07

P(X=x) = (nCx) px (1-p)n-x

Binomial Probabilities Table

X

P(X)

P(<=X)

P(<X)

P(>X)

P(>=X)

0

0.4840

0.4840

0.0000

0.5160

1.0000

1

0.3643

0.8483

0.4840

0.1517

0.5160

2

0.1234

0.9717

0.8483

0.0283

0.1517

3

0.0248

0.9964

0.9717

0.0036

0.0283

4

0.0033

0.9997

0.9964

0.0003

0.0036

5

0.0003

1.0000

0.9997

0.0000

0.0003

6

0.0000

1.0000

1.0000

0.0000

0.0000

7

0.0000

1.0000

1.0000

0.0000

0.0000

8

0.0000

1.0000

1.0000

0.0000

0.0000

9

0.0000

1.0000

1.0000

0.0000

0.0000

10

0.0000

1.0000

1.0000

0.0000

0.0000

Binomial Probabilities Table

X

P(X)

P(<=X)

P(<X)

P(>X)

P(>=X)

0

0.4840

0.4840

0.0000

0.5160

1.0000

1

0.3643

0.8483

0.4840

0.1517

0.5160

2

0.1234

0.9717

0.8483

0.0283

0.1517

3

0.0248

0.9964

0.9717

0.0036

0.0283

4

0.0033

0.9997

0.9964

0.0003

0.0036

5

0.0003

1.0000

0.9997

0.0000

0.0003

6

0.0000

1.0000

1.0000

0.0000

0.0000

7

0.0000

1.0000

1.0000

0.0000

0.0000

8

0.0000

1.0000

1.0000

0.0000

0.0000

9

0.0000

1.0000

1.0000

0.0000

0.0000

10

0.0000

1.0000

1.0000

0.0000

0.0000

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