Problem 1. In a family with 3 children, what is the probability of having 2 boys and 1 girls,?

Consider two cases:

a) Construct sample space where each element shows

sequence of given birth boys and girls, like Boy-Girl-Boy or Girl-Boy-Girl and so on.

Assuming that a boy is as likely as a girl at each birth, find probability that

b) 2 boys and 1 girl were born in that order;

c) 2 boys and 1 girl were born in any order.

Tip: Count in sample space number of events related to case b) and case c)

and divide by total number of all elements in the sample space.

Problem 2. Group of 10 people has 6 male and 4 female.

5 people were randomly selected from this group.

What is the probability that in selected people will be 3 male and 2 female?

Order of selection doesn’t matter.

Tip: calculate number of ways to select 3 Male out of 6, use Combinations C(6,3).

Calculate number of ways to select 2 Female out of 4, C(4,2).

Calculate number of ways to select any 5 People out of 10, C(10,5)

Probability will be: P = C(6,3)×C(4,2)/C(10,5)

Problem 3. What is the probability that a random card drawn from the full standard 52-card deck will be a Queen or any card of a Hearts? What are odds odds in favor for this event.

Problem 4. Use probabilities from the table below

B D

A 0.15 0.25

C 0.40 0.20

and formula for conditional probability to find the following probabilities:

a) P(A|B) probability of event A given that event B already happened;

b) P(B|A) probability of event B given that event A already happened.

Problem 5. Calculate Expected Value for the following Discrete Probability Distribution:

X -1 0 1 2 3 4

P(x) 0.04 0.08 0.12 0.32 0.24 0.16

Problem 6. Imagine you are playing a game by drawing a card

from a standard well-shuffled 52-cards deck.

Here are the rules:

If the card is an Ace, you win $5.

If the card is a King, you win $3.

If the card is a Queen, you win $3.

If the card is a Jack, you win $2.

If it’s any other card from 2 to 10, you lose $2.

After each draw card returns to the deck and it’s shuffled again.

So, every time you draw a card from the full deck of cards.

Complete the probability distribution table for this game:

Profit +$5 +$3 +$3 +$2 -$2

P(x)

Calculate the expected value of Profit in this game?

On a long run, will you win or lose in this game?

Problem 7. There are 20 tickets in the raffle box, 5 of them are winning.

You are randomly take two tickets from this box.

a) Here is a tree diagram for possible events.

Determine value for each of these probabilities: p1, p2, p3, p4, p5, p6.

Use these values to calculate probabilities to draw

b) two win tickets

c) one win and one lose ticket (in any order)

d) two lose tickets

Taken ticket is not returned to the box.

Problem 8. There are 2 candidates in the final election.

1,000 potential voters were asked if they like a certain candidate. Here is result of this poll: 300 people like candidate A, 400 people like candidate B, 100 people like both candidates (overlapping area between first and second group).

a) draw Venn diagram for this case;

b) use Venn diagram to find probability that people like candidate A, but don’t like candidate B;

c) find probability that people don’t like either of these candidates.

Problem 9. In a histogram, when the data are symmetrical, what is the typical relationship between the mean and median?

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