Probability with equally likely outcomes; the complement

About this lesson

Statistics (AP-aligned) › Probability

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Probability with equally likely outcomes; the complement

Warm-up

Three quick questions from earlier lessons.

Discover

Two dice are rolled. The sums go from 2 to 12: eleven values. Is each sum equally likely, with probability 111\tfrac{1}{11}?

  1. No
  2. Yes

Roll two dice 1,000 times. Which sum comes up most often?

  1. 7
  2. All sums come up about equally often

Key idea

Probability with equally likely outcomes, and the complement.

If all outcomes in the sample space are equally likely, P(A)=number of outcomes in Anumber of outcomesP(A) = \dfrac{\text{number of outcomes in } A}{\text{number of outcomes}}. Always 0≤P(A)≤10 \le P(A) \le 1, and P(not A)=1−P(A)P(\text{not } A) = 1 - P(A).

Worked example

At least one head in 3 coin tosses.

P(no heads)=18P(\text{no heads}) = \tfrac{1}{8}
8 equally likely sequences.
P(at least one head)=1−18P(\text{at least one head}) = 1 - \tfrac{1}{8}
Complement.
=78= \tfrac{7}{8}
Compute.

Your turn, guided

Your turn: P(sum = 7) with two dice.

16\tfrac{1}{6}
6 pairs of 36.

Practice ★

  1. Two fair dice are rolled. Find the probability that the sum is 2.
  2. Two fair dice are rolled. Find the probability that the sum is 12.
  1. The probability that a flight is late is 0.78. Find the probability that it is not late.

Watch out!

Practice ★★

  1. A fair die is rolled 2 times. Find the probability of at least one six.
  2. A fair die is rolled 6 times. Find the probability of at least one six.

Exit ticket

Three questions to check you have got it.

  1. Two fair dice are rolled. Find the probability that the sum is 2.
  2. Two fair dice are rolled. Find the probability that the sum is 12.
  3. Two fair dice are rolled. Find the probability that the sum is 8.

Summary card

Basics

  • Equally likely: favorable ÷ total.
  • P(not A)=1−P(A)P(\text{not } A) = 1 - P(A); "at least one" via the complement.

Watch out: make sure the outcomes really are equally likely.

End of the lesson

Next lesson: The addition rule and mutually exclusive events Back to the chapter

Answers: Probability with equally likely outcomes; the complement

A

  1. 136\frac{1}{36}
  2. 136\frac{1}{36}

B

  1. 1150\frac{11}{50}

C

  1. 1136\frac{11}{36}
  2. 3103146656\frac{31031}{46656}

D

  1. 136\frac{1}{36}
  2. 136\frac{1}{36}
  3. 536\frac{5}{36}

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