Probability Distribution Problems

**Page 1**

1.

There are 5 defective items in a batch of 25 items. 4 are chosen at random. Find the probability distribution of the defective items.

a. | II | ||

b. | IV | ||

c. | I | ||

d. | III |

Probability Distribution of defective items =

Correct answer : (3)

2.

Find the mean, variance and standard deviation for the number of heads, when 20 coins are tossed.

a. | 10, 5, $\sqrt{5}$ | ||

b. | 5, $\frac{\sqrt{5}}{2}$, $\frac{5}{2}$ | ||

c. | 10, $\sqrt{5}$, 5 | ||

d. | 5, $\frac{5}{2}$, $\frac{\sqrt{5}}{2}$ |

Mean = μ =

Variance =

Standard deviation =

So, the mean is 10, variance is 5, and standard deviation is

Correct answer : (1)

3.

If the probability of a defective bolt is 0.05, then find the mean and the standard deviation for the distribution of defective bolts in a total of 400.

a. | 19, $\sqrt{20}$ | ||

b. | 20, $\sqrt{19}$ | ||

c. | 19, 20 | ||

d. | 20, $\sqrt{20}$ |

Mean = μ =

Variance =

Standard deviation =

So, mean is 20, and standard deviation is

Correct answer : (2)

4.

The probability of a shooter hitting the target is $\frac{1}{3}$. If he tries 9 times, then what is the probability of his hitting the target at least two times?

a. | 11($\frac{{2}^{8}}{{3}^{9}}$) | ||

b. | 1- 11($\frac{{2}^{8}}{{3}^{9}}$) | ||

c. | 11($\frac{{2}^{9}}{{3}^{8}}$) | ||

d. | 1- 11($\frac{{2}^{9}}{{3}^{8}}$) |

Probability of hitting the target at least two times = P (

P(

P(

P (

Required probability = 1 - 11(

The probability of the shooter hitting the target at least two times is 1 - 11(

Correct answer : (2)

5.

Three defective pencils are mixed with 7 good ones. Two pencils are drawn at random simultaneously. Find the mean and standard deviation of the probability distribution of defective pencils drawn.

a. | 0.21, 1.212 | ||

b. | 0.6, 0.611 | ||

c. | 0.6, 0.422 | ||

d. | 0.42, 0.422 |

[Find the probability of drawing 0 defective pencils.]

P(X = 1) =

[Find the probability of drawing 1 defective pencil and 1 good pencil.]

P(X = 2) =

[Find the probability of drawing 2 defective pencils.]

The probability distribution for the defective pencils drawn is:

X | 0 | 1 | 2 |

P(X) |

The mean = μ =

The variance = σ

[Formula for the variance.]

σ

[Simplify.]

σ

[Simplify.]

Standard deviation σ =

[Simplify.]

So, the mean is 0.6, and the standard deviation is 0.611.

Correct answer : (2)

6.

6 dice are thrown 729 times. How many times do you expect at least 3 dice to show a 4 or 5?

a. | 323 | ||

b. | 322 | ||

c. | 233 | ||

d. | 232 |

Let the event be, getting 4 or 5 when a die is thrown.

Probability of occurrence of the event in a single throw is

Probability of occurrence of the event in at least 3 dice = P (E, 3) + P (E, 4) + P (E, 5) + P (E, 6)

This is a binomial situation and hence using the formula P (E,

The required probability = (6

Required probability = (

[Simplify.]

Out of 729 throws, the number of times atleast 3 dice will show a 4 or 5 is 729 ×

So, we can expect at least 3 dice to show a 4 or 5 for 233 times.

Correct answer : (3)

7.

a. | 75, 18.75 | ||

b. | 25, 12.25 | ||

c. | 75, 12.25 | ||

d. | 25, 18.75 |

Mean

Variance s

[Substitute

s

[Substitute

s

[Simplify.]

So, the mean is 25, and variance is 18.75 .

Correct answer : (4)

8.

a. | 165, 8.62 | ||

b. | 74.25, 8.62 | ||

c. | 165, 74.25 | ||

d. | 165, 12.85 |

Mean

[Substitute

Standard deviation s =

s =

[Substitute

s =

[Simplify.]

So, the mean is 165, and standard deviation is 8.62.

Correct answer : (1)

9.

30% of the people in a community went out on a vacation. If 600 people are selected at random, then find the standard deviation of the number of people who went out on vacation.

a. | 126 | ||

b. | 180 | ||

c. | 11.23 | ||

d. | 13.4 |

Probability that a person selected did not go on a vacation is

Number of people selected

Standard deviation

[Substitute

[Simplify.]

The standard deviation of the number of people who went out on vacation is 11.23 .

Correct answer : (3)

10.

In a school, 40% of the students play basketball. If 400 students are selected at random, then find the mean and variance of the number of students who play basketball.

a. | 160, 9.8 | ||

b. | 96, 160 | ||

c. | 96, 9.8 | ||

d. | 160, 96 |

Probability that a student selected does not play basketball is

Number of students selected

Mean

Variance s

[Substitute

s

[Substitute

So, the mean and variance of the number of students who play basketball is 160, 96 respectively.

Correct answer : (4)