Section 7B Combining Probabilities
And Probabilities
When finding the probability of event A and B occurring, the rule is to multiply individual probabilities. The
multiplication technique extends to situations involving two or more events occurring jointly. There are two
cases to consider, Independent and Dependent events.
Independent Events
Two events are independent if the outcome of one does not affect the probability of the other event. The
probability that A and B occur together is
P(A and B) = P(A) x P(B)
P(A and B and C)= P(A) x P(B) x P(C)
Example 1: Consecutive Floods
Find the probability that a 100-year flood (a flood with a 0.01 probability of striking in a given year) will strike a
city in two consecutive years. Assume that a flood in one year does not affect the likelihood of a flood in the
next year.
Example 2: Three Coins
Suppose you toss three coins. What is the probability of getting three tails?
Example 3: Rolling a Fair Die
Find the probability of rolling two 6s followed by one 5 on three tosses of a fair die.
Example 4: Standard deck of cards
Find the probability of drawing two hearts in a row from a standard deck of cards when the drawn card is
returned to the deck each time.
Dependent Events
Two events are dependent if the outcome of one affects the probability of the other event. The probability that
A and B occur together is
P(A and B) = P(A) x P(B given after A) where means “the probability of event B given the occurrence of event
A”
P(A and B and C)= P(A) x P(B given after A) x P(C given after A and B)
Example 5: Bingo
The game of Bingo involves drawing labeled buttons from a bin at random, without replacement. There are 75
buttons, 15 for each of the letters B, I, N, G, O. What is the probability of drawing two B buttons in the first
two selections?
Example 6: Jury Selection
A three-person jury must be selected at random from a pool of 12 people that has 6 men and 6 women. What
is the probability of selecting an all-male jury?
Example 7: Standard deck of cards
a) Find the probability of drawing two hearts in a row from a standard deck of cards when the drawn card is not
returned to the deck each time.
b) Find the probability of drawing a face card and then a 7 from a standard deck of cards when the drawn card
is not returned to the deck each time.
Either/Or Probabilities
When finding the probability that either of two events occur (rather than the probability of both events
occurring), the rule is to add individual probabilities. There are two cases to consider, Overlapping and Nonoverlapping (mutually exclusive) events.
Non-Overlapping Events (Mutually Exclusive)
When two events cannot occur at the same time, they are said to be non-overlapping events.
P(A or B)= P(A) + P(B)
P(A or B or C)= P(A) + P(B) + P(C)
Examples of non-overlapping events:
a) when rolling a fair die, you cannot land on a 6 and a 2 at the same time.
b) when drawing one card from a standard deck of cards, you cannot draw a face card and a 2 in the same
card
c) A coin cannot land on heads and tails at the same time
Example 8: Either/Or Dice
a) Suppose you roll a single die. What is the probability of rolling either a 2 or a 3?
b) Suppose you roll two dice. What is the probability of rolling a sum of either 4 or 9?
Example 9: Either/Or Standard deck of cards
Suppose you draw one card from a standard deck of cards. Find the probability of drawing either a face card or
a red 2.
Overlapping Events
Two events are overlapping if they can occur together.
P(A or B) = P(A) + P(B) – P(A and B)
Examples of non-overlapping events:
a) when rolling a fair die, you can land on a even number and a 6 at the same time
b) when drawing one card from a standard deck of cards, you can draw a face card and a red card at the same
time
Example 10: Either/Or Standard deck of cards (overlapping)
Suppose you draw one card from a standard deck of cards. Find the probability of drawing either a queen or a
club.
Example 11: Either/Or Dice (overlapping)
Suppose you roll a die on which the even numbers are red and the odd numbers are black. Find the probability
of rolling a 3 or a black number.
Example 12: Democrats and Women
You select one person at random from a room with eight people: two Democratic men, two Republican men,
two Democratic women, and two Republican women. What is the probability that you will select either a
woman or a Democrat?
The At Least Once Rule (For Independent Events)
Suppose the probability of an event A occurring in one trial is P(A). If all trials are independent, the probability
that event A occurs at least once in n trials is
P(at least one event A in n trials)= 1 – P(no events A in n trials)
= 1 – [P(not A in one trial)]n
Example 13: Toss a coin
Suppose you toss a coin four times. What is the probability of getting at least one head?
Example 14: 100-year flood
What is the probability that a region will experience at least one 100-year flood (a flood has a 0.01 chance of
occurring in any given year) during the next 100 years? Assume that 100-year floods in consecutive years are
independent events.
Example 15: Lottery Chances
You purchase 10 lottery tickets, for which the probability of winning some prize on a single ticket is 1 in 10.
What is the probability that you will have at least one winning ticket among the 10 tickets?
Example 16: Rolling a single die
Find the probability of rolling at least one 1 in six rolls of a single die.