Download Probability Review

IB Math HL – Lesson 24A-J
Probability Review
Name: ____________________________________ Block: ______
Experimental Probability:
Ex1) The table alongside shows the age distribution of prison inmates. Find the probability, as a percent to 3 sig figs, that:
a) a prisoner is male
c) a prisoner is 16 given that he is male
b) a prisoner is aged between 25 and 34
d) a prisoner is male given the prisoner is 16
e) a prisoner is no more than 29 given that the prisoner is female
Theoretical Probability:
Ex2) a) List the 8 possible 3-child families according to gender of the children. This is called the sample space.
b) Assuming that each of these is equally likely to occur, determine the probability that a family consists of:
i) all boys
ii) two girls and a boy
iii) a girl for the eldest
iv) at least one boy
Ex3) You toss a coin and spin the spinner shown.
a) Draw a grid to represent the sample space of possible outcomes.
b) Use your grid to determine the chance of getting:
i) a head and a number divisible by 3
iii) a number not divisible by 4
ii) a tail or an odd number
Ex4) People exiting a theme park ride are surveyed. Find the probability that a randomly selected person:
a) like the ride
b) is a child and disliked the ride
c) is an adult or disliked the ride
d) liked the ride, given that
he or she is a child
e) is an adult, given that he
or she disliked the ride
Independent events are events that do not influence each other. If A and B are independent events, then:
P(A and B) = P(A)
 P(B)
2-Dimensional Grids:
Ex5) You spin each of the spinners shown:
a) Draw a 2-dimensional grid to illustrate the sample space.
b) Find each of the following probabilities using the 2-dimensional grid AND using compound events.
i) a red and an odd number
ii) not yellow and a number divisible by 3
Ex6) A bag contains 5 blue marbles, 3 red marbles, 8 yellow marbles, and 4 green marbles. If you pick two marbles and replace
each marble after picking it, what is the probability of picking:
a) 2 blue marbles
b) a red, then a green
c) a red, then a yellow
d) 2 red
Dependent events are events for which the occurrence of one of the events does affect the others.
If A and B are dependent events, then:
P(A and B) = P(A)
 P(B given that A has occurred)
Ex7) A bag contains 5 blue marbles, 3 red marbles, 8 yellow marbles, and 4 green marbles. If you pick two marbles without
replacement, what is the probability of picking:
a) 2 blue marbles
b) a red, then a green
c) a red, then a yellow
d) 2 red
Ex8) A hat contains tickets with the numbers 1, 2, 3, …, 19, 20 on them. If three tickets are drawn from the hat, without
replacement, determine the probability that all are prime numbers.
Ex9) Two boxes each contain 6 petunia plants. Box A contains 2 purple and 4 white. Box B contains 5 purple and 1 white. A box is
selected by flipping a coin, and one plant is removed at random. Determine the probability it will have purple flowers.
Ex10) A bag contains 5 red and 3 blue marbles. Two marbles are drawn simultaneously from the bag. Determine the probability that
at least one of them is red.
Ex11) A box contains 3 red, 2 blue, and 1 yellow marble. Find the probability of getting two different colors:
a) if replacement occurs
b) if replacement does not occur
Ex12) Let U = {1, 2, 3, 4, 5, 6, 7} and A = {1, 3, 5} and B = {2, 4, 6}.
a) Fill in the information on the Venn diagram below:
b) Find A  B
Disjoint sets are sets which do not have elements in common.
They are also said to be mutually exclusive.
Ex13) If A is the set of all factors of 36 and B is the set of all factors of 54:
a) Write all the elements of sets A and B.
b) Find A  B
c) Find A  B
Ex14) The Venn Diagram below shows the number of people in a sporting club who play Tennis (T) and Hockey (H).
a) How many people are in the sporting club?
b) Determine the number of people:
i) who play both sports
ii) who play hockey
c) Determine the probability that a person chosen at random:
i) plays hockey, but not tennis
ii) plays tennis
iii) plays at least one sport
Ex15) In a class of 30 students, 19 study physics, 17 study chemistry, and 9 study neither.
a) Place the information in the Venn Diagram alongside.
b) Determine the probability that randomly selected a class member studies:
i) both subjects
ii) at least one of the subjects
iii) exactly one subject
iv) neither subject
v) chemistry, given that he/she studies physics
Addition Law of Probability:
P( A  B)  P( A)  P( B)  P( A  B)
Ex16) If P(A) = 0.6, P( A  B ) = 0.7, and P( A  B ) = 0.3, find P(B).
Addition Law of Probability, given that A and B are mutually exclusive:
P( A  B)  P( A)  P( B)
Ex17) A box of chocolates contains 6 with hard centers (H) and 12 with soft centers (S). Find:
a) P(H)
b) P(S)
c) P( H  S )
Conditional Probability:
P( A | B) 
P( A  B)
P( B)
d) P( H  S )
Ex18) The top shelf in a cupboard contains 3 cans of pumpkin soup and 2 cans of chicken soup. The bottom shelf contains 4 cans of
pumpkin soup and 1 can of chicken soup.
Lukas is twice as likely to take a can from the bottom shelf. If he takes one can of soup without looking at the label, find:
a) P(chicken)
b) P(top shelf | chicken)
Independent Events
Recall: If A and B are independent events, P ( A  B )  P ( A) P ( B ) . What about P ( A | B ) ?
Ex19) P ( A) 
1
1
, P ( B )  , and P ( A  B )  p . Find p if:
2
3
a) A and B are mutually exclusive
Ex20) Given P ( A) 
b) A and B are independent
1
1
2
, P ( B | A)  , and P ( B | A' )  ,
5
3
4
a) P ( B  A)
c) Fill in the information from (a)
and (b) on a Venn diagram
b) Find P ( B  A' )
d) Find P ( A  B ' )
e) Find P ( B )