Download Section 9-7 Probability of Mutliple Events

9-7
Probability of Multiple Events
Review-A bag contains 24 green marbles, 22 blue
marbles, 14 yellow marbles and 12 red
marbles. Suppose you pick one marble at
random. Find each probability.
7/36
P(yellow) =
P(not blue) =
25/36
P(green or red) = 1/2
• When the outcome of one event affects the
outcome of a second event, the two events are
dependent events.
• Example– Select a marble from a bag that
contains marbles of two colors. Put the marble
aside, and select a second marble from the bag.
• When the outcome of one event does not
affect the outcome of the second event, the
two events are independent events.
Examples of Independent events-• Examples—
Landing on heads after tossing a coin AND rolling a 5 on a single 6-sided die.
Choosing a marble from a jar AND landing on heads after tossing a coin.
Choosing a 3 from a deck of cards, replacing it, AND then choosing an ace as
the second card.
Rolling a 4 on a single 6-sided die, AND then rolling a 1 on second roll of the
die.
• To find the probability of two independent events
that occur in sequence, find the probability of each
event occurring separately, then multiply the
probabilities.
P(A and B) = P(A) * P(B)
• A dresser drawer contains one pair of socks with
each of the following colors: blue, brown, red, white
and black. Each pair is folded together in a
matching set. You reach into the sock drawer and
choose a pair of socks without looking. You replace
this pair and then choose another pair of socks.
What is the probability that you will choose the red
pair of socks both times?
P(red and red) =
P(red) * P(red) =
1/5 * 1/5 = 1/25
• A coin is tossed and a single 6 sided die is rolled.
Rind the probability of landing on the head side of
the coin and rolling a 3 on the die.
• P(head and 3) =P(head) * P(3) = 1/2* 1/6 = 1/12
Mutually Exclusive Events
• Two events are
mutually exclusive if
they cannot occur at
the same time (i.e.,
they have no outcomes
in common).
P(A)
P(B)
In the Venn Diagram above, the
probabilities of events A and B are
represented by two disjoint sets (i.e., they
have no elements in common).
• Two events are nonmutually exclusive if
they have one or more
outcomes in common.
P(A)
P(B)
In the Venn Diagram above, the
probabilities of events A and B are
represented by two intersecting sets (i.e.,
they have some elements in common).
Example-• A single card is chosen at random from a standard
deck of 52 playing cards. What is the probability of
choosing a 5 or a king?
• Possibilities—
The card chosen can be a 5.
The card chosen can be a king.
• Since they are mutually exclusive—
P(5 or king) = P(5) + P(king)= 4/52 + 4/52 = 8/52
Example-• A single 6-sided die is rolled. What is the
probability of rolling an odd number or an even
number?
• Possibilities—
The number rolled can be an odd number.
The number rolled can be an even number.
• Since they are mutually exclusive—
P(odd or even) = P(odd) + P(even)= 1/2 + 1/2 = 1
Example-• A single card is chosen at random from a standard
deck of 52 playing cards. What is the probability of
choosing a club or a king?
• Possibilities—
The card chosen can be a club.
The card chosen can be a king.
The card chosen can be a club and a king.
• Since they are NOT mutually exclusive—
P(Club or King) = P(Club) + P(king) – P(club and King)
= 13/52 + 4/52 – 52/2704
= 13/52 + 4/52 – 1/52
= 16/52 = 4/13
Example-• A single 6 sided die is rolled. What is the probability
of rolling a 5 or an odd number.
• Possibilities—
The number rolled can be a 5.
The number rolled can be an odd number.
The number rolled can be a 5 and an odd number.
• Since they are NOT mutually exclusive—
P(5 or odd) = P(5) + P(odd) – P(5 and odd)
= 1/6 + 3/6 – 3/6
= 1/6