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Thinking
Mathematically
Events Involving And; Conditional
Probability
Independent Events
Two events are independent events if the
occurrence of either of them has no effect
on the probability of the other.
And Probabilities with Independent
Events
If A and B are independent events, then
P(A and B) = P(A)•P(B)
Example Independent Events on a
Roulette Wheel
A roulette wheel has 38 numbered slots (1
through 36, 0, and 00). Of the 38
compartments, 18 are black, 18 are red, and
two are green. A play has the dealer spin the
wheel and a small ball in opposite
directions. As the ball slows to a stop, it can
land with equal probability on any one of
the 38 numbered slots. Find the probability
of red occurring on two consecutive plays.
Solution
The wheel has 38 equally likely outcomes and
18 are red. Thus the probability of red
occurring on a play is 18/38, or 9/19. The
result that occurs on each play is
independent of all previous results. Thus,
P(red and red) = P(red)•P(red) = 9/19•9/19
= 81/361
Example Hurricanes and Probability
If the probability that South Florida will be hit
by a hurricane in any single year is 5/19,
what is the probability that South Florida
will be hit by a hurricane in three
consecutive years?
Solution
The probability that South Florida will be hit
by a hurricane in three consecutive years is:
P(hurricane and hurricane and hurricane) =
P(hurricane)•P(hurricane)•P(hurricane) =
5/19•5/19•5/19 = 125/6859
Dependent Events
Two events are dependent events if the
occurrence of one of them has an effect on
the probability of the other.
And Probabilities with Dependent
Events
If A and B are dependent events, then
P(A and B) = P(A)•P(B given that A has occurred)
Example Dependent Events with Your
Cousins
Good news: You won a free trip to Madrid and can
take two people with you, all expenses paid. Bad
news: Ten of your cousins have appeared out of
nowhere and are begging you to take them. You
write each cousin’s name on a card, place the card
in a hat, and select one name. Then you select a
second name without replacing the first card. If
three of your ten cousins speak Spanish, find the
probability of selecting two Spanish-speaking
cousins.
Solution
Because P(A and B) = P(A)•P(B given that A has
occurred), then
P(two Spanish-speaking cousins) =
P(speaks Spanish)•P(speaks Spanish given that a
Spanish-speaking cousin was selected first)
= 3/10 • 2/9 = 6/90 = 1/15.
The probability of selecting two Spanish-speaking
cousins is 1/15.
Example An And Probability with Three
Dependent Events
Three people are randomly selected, one
person at a time, from five freshman, two
sophomores, and four juniors. Find the
probability that the first two people selected
are freshman and the third is a junior.
Solution
P(first two are freshman and the third is a junior)
= P(freshman)•P(freshman given that a freshman
was selected first)•P(junior given that a freshman
was select first and second)
= 5/11 • 4/10 • 4/9 = 8/99
The probability that the first two are freshman and
the third is a junior is 8/99.
Conditional Probability
The conditional probability of B, given A,
written P(B|A), is the probability that event
B occurs computed on the assumption that
event A occurs.
Example Computing Conditional
Probability
A letter is randomly selected from the letters
of the English alphabet. Find the probability
of selecting a vowel, given that the outcome
is a letter that precedes h.
Solution
Because we are given the condition that the
outcome is a letter that precedes h, the set of all
possible outcomes is
{a, b, c, d, e, f, g}
There are seven possible outcomes. We can select a
vowel from this set in one of two ways: a or e.
Therefore, the probability of selecting a vowel,
given that the outcome is a letter than precedes h,
is 2/7.
P(vowel | outcome precedes h) = 2/7
Applying Conditional Probability to
Real-World Data
P(B|A) =
observed number of times B and A occur together
observed number of times A occurs
Example Conditional Probabilities with
Real-World Data
The table shows the number of active-duty
personnel in the U.S. military in 2000. If one
person is randomly selected from the U.S.
military, find the probability that person is male if
the person is in the Marine Corps.
Army
Navy
Marine Corps Air Force
Male
402,602 316,858
161,571
288,271
Female 71,603
51,582
10,130
67,620
Solution
We find the probability that the selected person is
a male, given that we are selecting from the
Marines.
P(Male|Marine Corps) =
Observed number of people who are males and Marines
Observed number of people in the Marines
=(161,571)/(161,571+10,130) = 161,571/171,701
= 0.941
Thinking
Mathematically
Events Involving And; Conditional
Probability