L10.6 New

Algebra 2:

Algebra 2 Chapter 10 Lesson 6

Slide 2:

EXAMPLE 1 Construct a probability distribution Let X be a random variable that represents the sum when two six-sided dice are rolled. Make a table and a histogram showing the probability distribution for X . SOLUTION The possible values of X are the integers from 2 to 12 . The table shows how many outcomes of rolling two dice produce each value of X . Divide the number of outcomes for X by 36 to find P ( X ) .

Slide 3:

EXAMPLE 1 Construct a probability distribution

Slide 4:

EXAMPLE 2 Interpret a probability distribution Use the probability distribution in Example 1 to answer each question. What is the most likely sum when rolling two six-sided dice? What is the probability that the sum of the two dice is at least 10 ? SOLUTION The most likely sum when rolling two six-sided dice is the value of X for which P ( X ) is greatest. This probability is greatest for X = 7 . So, the most likely sum when rolling the two dice is 7 .

Slide 5:

EXAMPLE 2 Interpret a probability distribution The probability that the sum of the two dice is at least 10 is: P ( X > 10 ) = P ( X = 10) + P ( X = 11) + P ( X = 12) = 3 36 + 2 36 + 1 36 = 6 36 = 1 6 0.167

Slide 6:

GUIDED PRACTICE for Examples 1 and 2 A tetrahedral die has four sides numbered 1 through 4 . Let X be a random variable that represents the sum when two such dice are rolled. Make a table and a histogram showing the probability distribution for X .

Slide 7:

GUIDED PRACTICE for Examples 1 and 2 A tetrahedral die has four sides numbered 1 through 4 . Let X be a random variable that represents the sum when two such dice are rolled. What is the most likely sum when rolling the two dice? What is the probability that the sum of the two dice is at most 3 ? sum of 5; 3 16 ANSWER

Slide 8:

EXAMPLE 3 Construct a binomial distribution Sports Surveys According to a survey, about 41% of U.S. households have a soccer ball. Suppose you ask 6 randomly chosen U.S. households whether they have a soccer ball. Draw a histogram of the binomial distribution for your survey. SOLUTION The probability that a randomly selected household has a soccer ball is p = 0.41 . Because you survey 6 households, n = 6 .

Slide 9:

EXAMPLE 3 Construct a binomial distribution P ( k = 0) = 6 C 0 (0.41) 0 (0.59) 6 0.042 P ( k = 1) = 6 C 1 (0.41) 1 (0.59) 5 0.176 P ( k = 2) = 6 C 2 (0.41) 2 (0.59) 4 0.306 P ( k = 3) = 6 C 3 (0.41) 3 (0.59) 3 0.283 P ( k = 4) = 6 C 3 (0.41) 4 (0.59) 2 0.148 P ( k = 5) = 6 C 5 (0.41) 5 (0.59) 1 0.041 P ( k = 6) = 6 C 6 (0.41) 6 (0.59) 0 0.005

Slide 10:

EXAMPLE 3 Construct a binomial distribution A histogram of the distribution is shown.

Slide 11:

EXAMPLE 4 Interpret a binomial distribution Use the binomial distribution in Example 3 to answer each question. What is the most likely outcome of the survey? What is the probability that at most 2 households have a soccer ball? SOLUTION The most likely outcome of the survey is the value of k for which P ( k ) is greatest. This probability is greatest for k = 2 . So, the most likely outcome is that 2 of the 6 households have a soccer ball.

Slide 12:

EXAMPLE 4 Interpret a binomial distribution The probability that at most 2 households have a soccer ball is: P ( k < 2) = P ( k = 2) + P ( k = 1) + P ( k = 0) 0.306 + 0.176 + 0.042 0.524 ANSWER So, the probability is about 52% .

Slide 13:

GUIDED PRACTICE for Examples 3 and 4 In Sweden, 61% of households have a soccer ball. Suppose you ask 6 randomly chosen Swedish households whether they have a soccer ball. Draw a histogram showing the binomial distribution for your survey. ANSWER

Slide 14:

GUIDED PRACTICE for Examples 3 and 4 In Sweden, 61% of households have a soccer ball. Suppose you ask 6 randomly chosen Swedish households whether they have a soccer ball. 4 households; about 0.166 ANSWER

Slide 15:

EXAMPLE 5 Classify distributions as symmetric or skewed Describe the shape of the binomial distribution that shows the probability of exactly k successes in 8 trials if ( a ) p = 0.5 and ( b ) p = 0.9 . SOLUTION Symmetric; the left half is a mirror image of the right half.

Slide 16:

EXAMPLE 5 Classify distributions as symmetric or skewed Skewed; the distribution is not symmetric about any vertical line.

Slide 17:

GUIDED PRACTICE for Example 5 A binomial experiment consists of 5 trials with probability p of success on each trial. Describe the shape of the binomial distribution that shows the probability of exactly k successes if ( a ) p = 0.4 and ( b ) p = 0.5 ANSWER a. The distribution is skewed since it is not symmetric about a vertical line. b. The distribution is symmetric since it is symmetric about a vertical line.