Chapter 7 notes Binomial

Example Determine the probability of getting at least 14 heads in 20 tosses of a fair coin. Mean is = ?

Mean and Standard Deviation formulas n = the number of trials p = the probability of a success in one trial q = the probability of a failure in one trial

But… 1. There are n identical trials. 2. The n identical trials are independent. 3. The outcome for each trial can be classified as either a success or a failure. 4. The probability of success, p, is the same for each trial.

THIS IS BINOMIAL NOT NORMAL !!!! ?NORMALCDF?

Shade and label boundaries for “At least 14” 13.5 µ = 10 σ = 2.24 Alpha B

Test for approximation accuracy Therefore the answer is a Good APPROXIMATION

Exact Answer using the old method:

% Error by using the approximation

ENGLISH

Shade and label boundaries for “At least 14” 13.5

Shade and label boundaries for “At most 14” 14.5

Shade and label boundaries for “less than 14” 13.5

Shade and label boundaries for “greater than 14” 14.5

Shade and label boundaries for “exactly 14”

Shade and label boundaries for “between 8 and 24”

Shade and label boundaries for “outside 8 and 24”

Shade and label boundaries for “outside 8 and 24”

Shade and label boundaries for “outside 8 and 24” Create a second calculator method…

Shade and label boundaries for “outside 8 and 24”

Examples

Procedure for all examples 1) Shade a diagram and label the mean, raw scores, z scores and the percentages of areas. 2) Show work for the use of all formulas. 3) Generate raw scores and percentages to at least two (2) decimal places. 4) Generate probabilities to at least four (4)* decimal places.

= probability = raw score FINAL VERSION

1. There are n identical trials. 2. The n identical trials are independent. 3. The outcome for each trial can be classified as either a success or a failure. 4. The probability of success, p, is the same for each trial.

Mean and Standard Deviation Binomial formulas n = the number of trials p = the probability of a success in one trial q = the probability of a failure in one trial USE BOUNDARIES

For the experiment of tossing a coin 400 times, use the normal distribution to approximate the probability that: a) The coin lands heads at least 217 times.

Shade and label boundaries for “At least 217” Alpha B

b) The coin lands tails between 180 and 195 inclusive.

Shade and label boundaries for “between 180 and 195 inclusive”

c) The coin lands heads exactly 200 times.

Shade and label boundaries for “exactly 200”

d) The coin lands heads less than 185 or more than 210.

Shade and label boundaries for “less than 185 or more than 210”

A multiple choice test consists of 100 questions, each with five possible answers. If a student must answer at least 33 questions correctly in order to pass, approximate the probability that the student passes the test? (Assume the student guesses at all the answers). Redo with binomalCdf

Mean and Standard Deviation Binomial formulas

Shade and label boundaries for “At least 33” 32.5 Alpha B µ = 20 σ = 4

Medical records show that 25% of all people suffering from “hardening of the arteries” have side effects from a certain type of medication. If 200 people suffering from “hardening of the arteries” are given this medication, what is the probability that:

a) No more than 35 will have side effects?

Mean and Standard Deviation Binomial formulas

Shade and label boundaries for “At most 35” 35.5 µ = 50 σ = 6.12 Alpha A

b) At least 160 will not have side effects?

At least 160 will not have side effects is at most 40 will.

At least 160 will Not = Wi ll

Shade and label boundaries for “At most 40” 40.5 µ = 50 σ = 6.12 Alpha A

a) No more than 35 will have side effects? (N=200 was done) Redo part ”a” for 300 people instead of 200. Redo part ”a” for 400 people instead of 200. Compare the answers for 200, 300 and 400. What do you notice?

Mean and Standard Deviation Binomial formulas

Shade and label boundaries for “At most 35” 35.5 µ = 75 σ = 7.5 Alpha A

Mean and Standard Deviation Binomial formulas

Shade and label boundaries for “At most 35” 35.5 µ = 100 σ = 8.66 Alpha A

MORE EXAMPLES

The probability that a camcorder battery lasts three years is 0.80.What is the probability that between 330 and 340 of the next 400 batteries sold last three years?

Mean and Standard Deviation Binomial formulas

Shade and label boundaries for “between 330 and 340” µ = 320 σ = 8

An airline company knows that 90% of the people making flight reservations for a certain flight will show up for the flight. What is the probability that out of the next 10,000 people making flight reservations:

Mean and standard deviation

a) 9075 or less will show up for the flight? b) At least 9030 of them will show up for the flight? c) Exactly 1000 will not show up for the flight?

Shade and label boundaries for “At most 9075” Alpha A

b) At least

c) “exactly 1000 will not”

c) “exactly 1000 will not” NOT CORRECT

c) “exactly 1000 will not” is “9000 will”

From past experience a restaurant owner knows that 80% of the drinks ordered will be alcoholic. For the next 400 drinks ordered: a) How many would be expected to be alcoholic? b) What is the probability that at least 300 drinks will be alcoholic?

a) Mean

b) What is the probability that at least 300 drinks will be alcoholic?

Standard Deviation

Shade and label boundaries for “At least 300” µ = 320 σ = 8 Alpha B

On a good weekend the owner sells at least $990 worth of alcoholic drinks at $3.00 per drink. c) If 400 drinks are ordered next weekend what is the probability that next weekend is good?

A good weekend

b) At least Alpha B

d) The owner estimates that 25,600 drinks will be sold during the next year. He knows that for every 30 alcoholic drinks sold he uses one bottle of liquor. If each bottle of liquor costs him $9.00, how much money should he budget to be 95% confident his liquor costs are covered?

Medical researchers at a Boston university report that 20% of babies born to American mothers this year will be delivered by cesarean sections. What is the probability that out of the next 300 deliveries:

a) Not more than 85 will be cesarean sections? b) At least 260 will not be cesarean sections?

Mean and Standard Deviation Binomial formulas

Shade and label boundaries for “ a) Not more than 85 ” 85.5 Alpha A

b) At least 260 will not be cesarean sections? is At most 40 will

Shade and label boundaries for “At most 40 will” 40.5 Alpha A

END

Procedure for all examples 1) Shade a diagram and label the mean, raw scores, z scores and the percentages of areas. 2) Show work for the use of all formulas. 3) Generate raw scores and percentages to at least two (2) decimal places. 4) Generate probabilities to at least four (4)* decimal places.

= probability = raw score FINAL VERSION Raw scores have labels USE BOUNDARIES in Binomials

BINOMIAL IS ALWAYS normalCdf… 1. There are n identical trials. 2. The n identical trials are independent. 3. The outcome for each trial can be classified as either a success or a failure. 4. The probability of success, p, is the same for each trial.

Mean and Standard Deviation Binomial formulas n = the number of trials p = the probability of a success in one trial q = the probability of a failure in one trial USE BOUNDARIES

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