High School

A large high school offers AP Statistics and AP Calculus. Among the seniors in this school, 65% take AP Statistics, 45% take AP Calculus, and 30% take both.

If a senior is randomly selected, what is the probability that they are in AP Statistics or AP Calculus but not both?

Answer :

The probability that a randomly selected senior is in AP Statistics or AP Calculus but not both is 0.50

We need to find the probability that a randomly selected senior at a large high school is in AP Statistics or AP Calculus but not both.

Given Information:

  1. 65% of seniors take AP Statistics: [tex]P(\text{Statistics}) = 0.65[/tex]
  2. 45% of seniors take AP Calculus: [tex]P(\text{Calculus}) = 0.45[/tex]
  3. 30% of seniors take both classes: [tex]P(\text{Both}) = 0.30[/tex]

  1. Calculate the probability of a senior taking either AP Statistics or AP Calculus:

    We use the formula for the union of two sets:
    [tex]P(\text{Statistics or Calculus}) = P(\text{Statistics}) + P(\text{Calculus}) - P(\text{Both})[/tex]

    Substituting in the given values:
    [tex]P(\text{Statistics or Calculus}) = 0.65 + 0.45 - 0.30 = 0.80[/tex]

    So, the probability that a senior is taking either AP Statistics or AP Calculus is [tex]0.80[/tex].

  2. Calculate the probability of a senior taking either AP Statistics or AP Calculus but not both:

    To find the probability that a senior is taking either AP Statistics or AP Calculus but not both, we need to subtract the probability of taking both from the probability of taking either one of the classes:
    [tex]P(\text{Statistics or Calculus but not Both}) = P(\text{Statistics or Calculus}) - P(\text{Both})[/tex]

    Substituting the calculated value and the given value:
    [tex]P(\text{Statistics or Calculus but not Both}) = 0.80 - 0.30 = 0.50[/tex]

    So, the probability that a senior is taking either AP Statistics or AP Calculus but not both is [tex]0.50[/tex].

The probability that a senior is in either AP Statistics or AP Calculus, but not both, is 0.50 or 50%.

To find the probability that a senior is in AP Statistics or AP Calculus but not both, we need to use the principle of inclusion-exclusion.

Let's define:

- ( S ) as the event that a senior takes AP Statistics,

- ( C ) as the event that a senior takes AP Calculus.

We are given:

- ( P(S) = 0.65 ) (65% take AP Statistics),

- ( P(C) = 0.45 ) (45% take AP Calculus),

[tex]- \( P(S \cap C) = 0.30 \)[/tex] (30% take both AP Statistics and AP Calculus).

We want to find[tex]\( P((S \cup C) \cap \neg(S \cap C)) \), where \( \neg \)[/tex]denotes "not".

Using the inclusion-exclusion principle:

[tex]\[ P((S \cup C) \cap \neg(S \cap C)) = P(S) + P(C) - 2P(S \cap C) \][/tex]

Let's calculate this:

[tex]\[ P((S \cup C) \cap \neg(S \cap C)) = P(S) + P(C) - 2P(S \cap C) \][/tex]

[ = 0.65 + 0.45 - 2(0.30) ]

[ = 0.65 + 0.45 - 0.60 ]

[ = 0.50 ]

So, the probability that a senior is in AP Statistics or AP Calculus but not both is ( 0.50 ), or 50%.

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