Theoretical or Experimental

1. Maria flipped a coin and got 6 heads out of 10 flips.

2. Carlos said the chances of rain today are 30%.

3. James said he has a 70% chance of making a free throw because yesterday he made 7 out of 10.

4. 6 students out of 18 students in one classroom caught a cold, so the nurse said about 33% of the students in school would catch the same cold.

5. Julia placed an eraser under one of four cups and told Patrick he had a 25% chance of finding the correct cup.

Compound Events – Part 1 IndependentWe will explore the calculation of the probability for compound events, which are those involve two or more outcomes.

P(E) = P of E;

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P(E) = P of E bar

P(E) represents the probability that an event E will occur.

represents the probability that an event E will not occur.

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P(E)

€

TheE .is called the complement of event E

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P(E) +P(E) =1

The complement of event E is “when anything else happens except for E.”

In tossing a penny, a nickel, and a dime once, what is the probability that the coins do not all land on tails?

If E represents the event that all three coins land on tails.

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( )Then P E=18; (PE) =

78

Asking 90 people to choose their favorite ice cream flavor.

FlavorFrequen

cy

vanilla 32

chocolate 24

strawberry

16

butter pecan

10

cherry 8

What is the probability that a randomly selected person does not choose vanilla or chocolate as his/her favorite flavor?Let F represent the event of selecting a person who chooses vanilla or chocolate

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P(F ) =5690

=2845

then, (PF) =1-2845

=1745

≈37.8%

Math Flash!!!Another way to approach the

solution is to let F represent the event of selecting a person whose favorite flavor is neither vanilla nor chocolate, for which there are 34 people.

Then, the required probability is

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P(F ) =3490

=1745

≈37.8%

Independent

Events A and B are called independent if the probability of one event to occur has no effect on the probability of the other event to occur.

For example:Event A could represent “rain for

tomorrow”Event B could represent “John is

feeling well today”Event A and event B are

independent from one another. We can safely assume that the probability of rain tomorrow has no effect on the probability of how John is feeling today.

Are these events independent?A card is chosen randomly from a

standard deck of cards, recorded, and then replaced. A second card is then drawn and recorded. Consider the events below.

A= first card is an ace of spadesB= second card is an ace of

spades

Independent Compound EventThe Multiplication Rule for Probability

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P A∩B( ) =P(A) ×P(B)

Events A and B occurred, and independent

For example:Suppose a penny is tossed once and

a nickel is tossed once. What is the probability that the penny lands on tails and the nickel lands on heads?

Let A = the penny lands on tailsLet B = the nickel lands on heads

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( )Then P A=12 ( )and P B=

12,

(so P A∩ )B =12×12=14=25%

For example:An ordinary dime is tossed twice, and

a die is rolled once. What is the probability that the dime will show tails both times and the die will show a 5?

Let E = the dime shows tail both timesLet F = the die shows a 5

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( )P E =14 ( )and P F=

16,

(then P E∩ )F =14×16=

124

≈4.17%

Another example:In drawing two cards from a deck of

cards, one at a time, with replacement of the first card, prior to drawing the second card, what is the probability of drawing 2 clubs?

Let G = drawing a club for the first card

Let H = drawing a club for the second card

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( )Since P G= ( )P H =1352

=14,

P(G∩ H ) =14×14=

116

=6.25%

Another Example:One card is drawn from a deck of cards,

and nickel is tossed 3 times. What is the probability that a picture card will be drawn and that the nickel will land on tails exactly twice?

Let M = drawing a picture cardLet N = getting exactly 2 tails

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P(M ∩ N ) =313

×38=

9104

≈8.65%

Another example:In a bag of jelly beans there are 6

red, 8 yellow, 4 green, and 7 black ones. Two jelly beans will be randomly selected, one at a time, with replacement. What is the probability of drawing a red jelly bean, followed by a black jelly bean?

Let S = drawing a red jelly beanLet T = drawing a black jelly bean

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P(S∩T) =625

×725

=42625

=6.72%

A person will be randomly selected from this group of 90 people, and a die is roll twice. What is the probability of selecting a person whose favorite ice cream flavor is strawberry, and rolling a sum of 10?

FlavorFrequen

cy

vanilla 32

chocolate 24

strawberry

16

butter pecan

10

cherry 8

Let V = the first eventLet W = the second event.

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P V ∩W( ) =845

×112

=8

540=

2135

The probability of selecting a person whose favorite ice cream flavor is strawberry, and rolling a sum of 10 is 1.48%

The results of rolling a weighted die 80 times are as follows:Outcom

eFrequen

cy

1 15

2 12

3 10

4 20

5 17

6 6

If the die is rolled twice, what is the probability that it will land on a number less than 3 both times?Let x = first rollLet y = second roll

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P(X ∩Y ) =2780

2780

⎛

⎝ ⎜

⎞

⎠ ⎟=

7296400

The probability that the die will land on a number less than 3 both times is 11.4%.

True or False

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P(A∩B) =P(B∩ A)For example: from the previous jelly beans problemLet S = drawing a red jelly beanLet T = drawing a black jelly bean

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P(S∩T) =625

×725

=42625

=6.72%

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P(T ∩U ) =725

×625

=42625

=6.72%