MULTIPLICATION RULES FOR PROBABILITY

INDEPENDENT AND DEPENDENT EVENTS

When do you multiply?

• Multiplication should be used whenever you have multiple (compound) events.

• Compound events can either be Independent or Dependent.

Independent Events

• Two events are Independent when the probability of one event occurring does not affect the probability of the other event.

• Examples of independent events:• Rolling a die/ tossing a coin• Tossing two coins• Spinning a number spinner multiple times

Multiplication Rule for Independent Events

• When 2 events are independent, the probability of both A and B occurring is:

• P(A and B) = P(A)·P(B)• Example 1: Joe tosses a coin and then rolls a

die. Find the probability of getting a head on the coin and a 4 on the die.

• P(H and 4) = P(H)·P(4)

1 1 1(H and 4)

2 6 12P

Ex 2: Independent Events

• Joe tosses 2 coins. Find the probability that he gets a head on one coin and a tail on the other.

• Solution: 1st make sure you are considering all possiblities in the sample space. This can easily be accomplished with either a table or a tree diagram

Table for Tossing 2 Coins

Heads Tails

Heads H, H H, T

Tails T, H T, T

Sample Space: {HH, HT, TH, TT}

Back

Tree Diagram for Tossing 2 Coins

Heads

Heads

Heads

Tails

Tails

Tails

Sample Space:

{HH, HT, TH, TT}

Solution

Ex 2: Solution

By formula:

P(H and T) = P(H and T) or P(T and H)

= .5·.5 + .5·.5

=.25 + .25 = .50

Or use sample space: {HH, HT, TH, TT}

Ex 3: Independent Events• A poll found that 38% of Americans say they

suffer great stress at least once per week. If 3 people are selected at random, find the probability that all 3 will say they suffer great stress at least once per week.

• Solution: Choosing 3 people from all Americans is not going to affect the sample space – referred to as the law of large numbers

P(S and S and S) = P(S)·P(S)·P(S)

=.38·.38·.38 = .055

Example 3: extended

• If 3 people are selected at random, find the probability that none will suffer great stress at least once per week.

=.62·.62·.62 = .238

Dependent Events

• Two events are dependent when the probability of one event occurring does affect the probability of the second event.

• Examples of dependent events:• Choosing 2 cards from a deck of cards without

replacing the first one.• Selecting a ball from a bag without replacement• Selecting multiple students from a small class

Multiplication Rule for Dependent Events

• When 2 events are dependent, the probability of both A and B occurring is:

• P(A and B) = P(A)·P(B|A)

The probability of B occurring given that A has already occured

Ex 1: Dependent Events

• A bag contains 2 red, 5 blue and 3 green marbles. 2 marbles are drawn one at a time without replacing the first one. Find the following probabilities:

a) Of selecting 2 blue marbles

b) Of selecting 1 blue and 1 red marble.

Ex 1: Solution (Dependent)

Construct a tree diagram

R = 2/10

R

R = 2/9

R

B = 5/10

B

B = 4/9

B = 5/9

G = 3/10

G

G

G

a) P(B and B) = (5/10) ·(4/9) = 20/90

b) P(B and R) = (5/10)·(2/9) + (2/10)·(5/9) = 20/90

Ex 1: Discussion

1. How would the problem change if you replaced the first marble?

2. How would the original problem change if you would have specified order of selection in part b?

Ex 2: Dependent/Independent

• Find the probability that 2 people selected at random are

a) Both born in the summer?b) Born in the same month in the summer?

Solution: a) P(S and S) = P(S)·P(S) =(3/12)·(3/12) =

1/16 b) P(S and Same) =P(S)·P(Same) =(3/12)·(1/12) = 1/48