Chapter 5: Division and Proportions in Algebra Lesson 6: Probability Distributions Mrs. Parziale.

Chapter 5: Division and Proportions in Algebra

Lesson 6: Probability Distributions

Mrs. Parziale

How are probabilities determined

• Probabilities are determined in one of three ways:– Use the relative frequency– Deduce from assumptions about a situation– Give a best guess

Vocabulary

• Outcome – Result of an ______________.• Event – Set of ______________________.• Probability – A number from ___ to ___ that

measures the likelihood that an event will occur.

• Probability Distribution – The set of _______________ of outcomes and their probabilities.

experimentpossible outcomes

0

ordered pairs

1

Example 1:

According to the National Center of Health Statistics, there were 4,089,950 children born in 2003 in the US. The total number of boys born was 2,093,535. What could be the probability a boy would be born that year? Describe the probability using the three ways noted above.

• Using the relative frequency –

• Deduce from assumptions –

• Best guess -

Rules or Properties:

Example 2:

A standard deck of playing cards has 52 cards. There are four suits: clubs , diamonds , hearts , and spades . Each suit has 13 cards: ace, 2,3,4,5,6,7,8,9,10, jack, queen, and king. Suppose you shuffle the cards well and pull one card at random. Find:

a) P(ace of hearts)

b) P(queen)

c) P(not a queen)

Probability Distributions: • Some situations have many outcomes. A probability distribution is the set of

ordered pairs of outcomes and their probabilities. For example, in many board games, two dice are thrown and the spaces moved depend on the sum of the two dice. The chart below shows the likelihood of sums of two dice.

• How many possible outcomes are there of getting a sum of two dice? ________________

• Complete the table to find the total number of possible sums.Sum Total (x)

Number of times the sum occurs

Probability the sum occurs

Decimal value

Sum Total (x)

Number of times the sum occurs

Probability the sum occurs

Decimal value

1 7

2 8

3 9

4 10

5 11

6 12

What’s the Probability?• What is the probability getting a sum of 1? P(1) = _________________

• What is the probability of getting a sum of 4? P(4) = _______________

• Find P(7) = _________________Find P(12) = ____________________

• Find the probability of tossing a sum greater than 8? _________________

Complements

Example 4: A weather forecaster reports that the probability of rain tomorrow is 70%. What is the probability that it does not rain?

P(E) + P(complement of E) = _____________

P(complement of E) = _____________

Example 5:

• If a number is randomly picked from {11, 12, 13, 14, 15, 16, 17}, what is the probability that it is divisible by 3? What is the probability the number picked is not divisible by 3?

Odds:

Probabilities and complements of probabilities are used to compute odds. Odds of an event occurring can be found by dividing the probability of an event occurring by the probability it will not occur.

Example 6: Suppose the probability that an event

(getting a sum of 7 when throwing 2 dice) is ,

what are the odds in favor of getting a sum of 7.

( ) ( )

( ) 1 ( )

P E P Eodds of event E occurring =

P complement of E P E

1

6

Example 7:

• Suppose the probability of an event will occur is .

• What are the odds in favor of the event occurring?

• What are the odds in favor of the event not occurring?

5

12

Closure

• What is an outcome?• What is an event?• Probability ranges between ___ and ____.• If the probability of an event is 3/5, what is

the complement?• What are the odds of such an event?