7.2 Multistage Experiments with Tree Diagrams and Geometric Probability

Tree Diagrams can be very useful.

With Replacement versus Without Replacement

Draw two balls from an Urn with 2 Red and 3 Green

A) With Replacement B) Without Replacement

Multiplication Rule for Probability

For all multistage experiments, the probability of the outcome along any path of a tree is equal to the product of the probabilities along the path.

Sample spaces are the same – but outcomes are NOT equally likely

S = {RR, RG, GR, GG}

With Replacement Without Replacement

Probability	RR	RG	GR	GG	Probability	RR	RG	GR	GG
With					Without

Find the Probability of drawing one of each color

P(with replacement)

P(without replacement

Find Probability of drawing at least one red with and without

An executive committee consisted of ten members: four women and six men. Three members were selected at random to be sent to a meeting in Hawaii. A blindfolded woman drew three of the ten names from a hat. All three names drawn were women’s. What was the probability of such luck?

Draw a tree diagram

Following are three boxes containing letters:

MATH	AND	HISTORY
1	2	3

If a box is chosen at random and then a letter is drawn at random from the box, what is the probability that the letter is an H? Another tree diagram.

MATH	AND	HISTORY
1	2	3

Variation: One letter is drawn at random from box 1, then another from box 2, then another from box 3, with the results recorded in order. What is the probability that the outcome is MAT?

Consider a box containing letters of word |DEPENDABLE|

If draw one letter at a time without replacement until have 4 letters, what is the probability spelled out NEED.

"just use one branch" of tree diagram

· N · E · E · D ·

P(NEED) =

P(LEAD)

An assembly line has two inspectors. The probability that the first inspector will miss a defective item is 0.05. If the defective item passes the first inspector, the probability that the second inspector will miss it is 0.01. What is the probability that a defective item will pass by both inspectors?

Complements

Often useful for “at least one” or situations where easier to find probability of the complement of what is really desired.

A husband and wife discover that there is a 10% probability of their passing on a hereditary disease to one of their children. If they plan to have three children, what is the probability that at least one child will inherit the disease?

What is the complement of “at least one child”?

Consider the experiment:

O X X	O O X	X O O
1	2	3

Draw a letter from box 1, put it in box 2, then draw from 2 put in 3, then draw from three. What is the probability that the last one drawn is an X.

Versus First choose a box, then draw letter. What is the probability of drawing an X.?

Independent Events

The outcome of one event has no effect on the outcome of a second event.

If events E and F are independent, then

“with replacement” is independent

A box contains 3 white balls and 2 black balls m m m l l. A ball is drawn at random from the box, its color is recorded, and then it is put back in the box and a second ball is drawn from the box. Find the probability that the two balls are of different colors.

Geometric Probability

Probability shaded =

More Practice

A nickel, a dime and a quarter are tossed. What is the probability of obtaining at least three heads?

Brittany is going to ascend a four-step staircase. At any time, she is just as likely to stride up one step or two steps. Find the probability that she will ascent the four steps in

a) two strides

b) three strides

c) four strides