Thinking Mathematically

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Thinking Mathematically

• Events Involving And Conditional Probability

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Independent Events

• Two events are independent events if the
  occurrence of either of them has no effect on the
  probability of the other.

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And Probabilities with Independent Events

• If A and B are independent events, then
• P(A and B) P(A)P(B)

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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.

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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)
  9/199/19 81/361

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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?

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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)
• 5/195/195/19 125/6859

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Dependent Events

• Two events are dependent events if the
  occurrence of one of them has an effect on the
  probability of the other.

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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)

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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 cousins 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.

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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.

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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.

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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.

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Conditional Probability

• The conditional probability of B, given A,
  written P(BA), is the probability that event B
  occurs computed on the assumption that event A
  occurs.

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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.

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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

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Applying Conditional Probability to Real-World
Data

• P(BA)
• observed number of times B and A occur together
• observed number of times A occurs

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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.

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Solution

• We find the probability that the selected person
  is a male, given that we are selecting from the
  Marines.
• P(MaleMarine Corps)
• Observed number of people who are males and
  Marines
• Observed number of people in the Marines
• (161,571)/(161,57110,130) 161,571/171,701
  0.941

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Thinking Mathematically

• Events Involving And Conditional Probability