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In a standard deck, there are 52 cards. Twelve cards are face cards ( F ) and 40 cards are not face cards ( N ). Draw two cards, one at a time, without replacement. The tree diagram is labeled with all possible probabilities.

This is a tree diagram with branches showing frequencies of each draw. The first branch shows 2 lines: F 12/52 and N 40/52. The second branch has a set of 2 lines (F 11/52 and N 40/51) for each line of the first branch. Multiply along each line to find FF 121/2652, FN 480/2652, NF 480/2652, and NN 1560/2652.
  1. Find P ( FN OR NF ).
  2. Find P ( N | F ).
  3. Find P (at most one face card).
    Hint: "At most one face card" means zero or one face card.
  4. Find P (at least on face card).
    Hint: "At least one face card" means one or two face cards.
  1. P ( FN OR NF ) = 480 2,652  +  480 2,652  =  960 2,652  =  80 221
  2. P ( N | F ) = 40 51
  3. P (at most one face card) = (480  +  480  +  1,560) 2,652 = 2 , 520 2 , 652
  4. P (at least one face card) = (132 + 480 + 480) 2,652 = 1,092 2,652

A litter of kittens available for adoption at the Humane Society has four tabby kittens and five black kittens. A family comes in and randomly selects two kittens (without replacement) for adoption.

This is a tree diagram with branches showing probabilities of kitten choices. The first branch shows two lines: T 4/9 and B 5/9. The second branch has a set of 2 lines for each first branch line. Below T 4/9 are T 3/8 and B 5/8. Below B 5/9 are T 4/8 and B 4/8. Multiply along each line to find probabilities of possible combinations.

  1. What is the probability that both kittens are tabby?
    a. ( 1 2 ) ( 1 2 ) b. ( 4 9 ) ( 4 9 ) c. ( 4 9 ) ( 3 8 ) d. ( 4 9 ) ( 5 9 )
  2. What is the probability that one kitten of each coloring is selected?
    a. ( 4 9 ) ( 5 9 ) b. ( 4 9 ) ( 5 8 ) c. ( 4 9 ) ( 5 9 ) + ( 5 9 ) ( 4 9 ) d. ( 4 9 ) ( 5 8 ) + ( 5 9 ) ( 4 8 )
  3. What is the probability that a tabby is chosen as the second kitten when a black kitten was chosen as the first?
  4. What is the probability of choosing two kittens of the same color?

a. c, b. d, c. 4 8 , d. 32 72

Try it

Suppose there are four red balls and three yellow balls in a box. Three balls are drawn from the box without replacement. What is the probability that one ball of each coloring is selected?

( 4 7 ) ( 3 6 ) + ( 3 7 ) ( 4 6 )

Venn diagram

A Venn diagram is a picture that represents the outcomes of an experiment. It generally consists of a box that represents the sample space S together with circles or ovals. The circles or ovals represent events.

Suppose an experiment has the outcomes 1, 2, 3, ... , 12 where each outcome has an equal chance of occurring. Let event A = {1, 2, 3, 4, 5, 6} and event B = {6, 7, 8, 9}. Then A AND B = {6} and A  OR  B = {1, 2, 3, 4, 5, 6, 7, 8, 9}. The Venn diagram is as follows:

A Venn diagram. An oval representing set A contains the values 1, 2, 3, 4, 5, and 6. An oval representing set B also contains the 6, along with 7, 8, and 9. The values 10, 11, and 12 are present but not contained in either set.

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Suppose an experiment has outcomes black, white, red, orange, yellow, green, blue, and purple, where each outcome has an equal chance of occurring. Let event C = {green, blue, purple} and event P = {red, yellow, blue}. Then C AND P = {blue} and C OR P = {green, blue, purple, red, yellow}. Draw a Venn diagram representing this situation.

Flip two fair coins. Let A = tails on the first coin. Let B = tails on the second coin. Then A = { TT , TH } and B = { TT , HT }. Therefore, A AND B = { TT }. A OR B = { TH , TT , HT }.

The sample space when you flip two fair coins is X = { HH , HT , TH , TT }. The outcome HH is in NEITHER A NOR B . The Venn diagram is as follows:

This is a venn diagram. An oval representing set A contains Tails + Heads and Tails + Tails. An oval representing set B also contains Tails + Tails, along with Heads + Tails. The universe S contains Heads + Heads, but this value is not contained in either set A or B.

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Roll a fair, six-sided die. Let A = a prime number of dots is rolled. Let B = an odd number of dots is rolled. Then A = {2, 3, 5} and B = {1, 3, 5}. Therefore, A AND B = {3, 5}. A OR B = {1, 2, 3, 5}. The sample space for rolling a fair die is S = {1, 2, 3, 4, 5, 6}. Draw a Venn diagram representing this situation.

Forty percent of the students at a local college belong to a club and 50% work part time. Five percent of the students work part time and belong to a club. Draw a Venn diagram showing the relationships. Let C = student belongs to a club and PT = student works part time.

This is a venn diagram with one set containing students in clubs and another set containing students working  part-time. Both sets share students who are members of clubs and also work part-time. The universe is labeled S.

If a student is selected at random, find

  • the probability that the student belongs to a club. P ( C ) = 0.40
  • the probability that the student works part time. P ( PT ) = 0.50
  • the probability that the student belongs to a club AND works part time. P ( C AND PT ) = 0.05
  • the probability that the student belongs to a club given that the student works part time. P ( C | P T )   =   P ( C  AND  P T ) P ( P T )   =   0.05 0.50   =   0.1
  • the probability that the student belongs to a club OR works part time. P ( C OR PT ) = P ( C ) + P ( PT ) - P ( C AND PT ) = 0.40 + 0.50 - 0.05 = 0.85

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Source:  OpenStax, Introduction to statistics i - stat 213 - university of calgary - ver2015revb. OpenStax CNX. Oct 21, 2015 Download for free at http://legacy.cnx.org/content/col11874/1.3
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