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ProbabilityStatistics and probabilityHigh schoolabout 0:58

Independent events have P(A) = 0.5 and P(B) = 1/3. Find P(A and B) and P(A or B): animated lesson

A narrated, step-by-step probability video lesson with captions. Final answer: P(A and B) = 1/6 ≈ 0.16666667; P(A or B) = 2/3 ≈ 0.66666667; P(neither) = 1/3 ≈ 0.33333333.

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Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.4 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Worked example
Independent events have P(A) = 0.5 and P(B) = 1/3. Find P(A and B) and P(A or B).
2. Step by step
1. $P(A \cap B) = P(A)P(B) = (\frac{1}{2})(\frac{1}{3}) = \frac{1}{6} \approx 0.16666667$
  Both: multiply, because the events are independent
2. $P(A \cup B) = \frac{1}{2} + \frac{1}{3} - \frac{1}{6} = \frac{2}{3} \approx 0.66666667$
  At least one: add and subtract the overlap
3. $P(A \cup B) - P(A \cap B) = \frac{1}{2} \approx 0.5$
  Exactly one of them
4. $(1 - P(A))(1 - P(B)) = \frac{1}{3} \approx 0.33333333$
  Neither
3. The answer
$P(A\cap B) = \frac{1}{6} \approx 0.16666667,\ P(A \cup B) = \frac{2}{3} \approx 0.66666667$
P(A and B) = 1/6 ≈ 0.16666667; P(A or B) = 2/3 ≈ 0.66666667; P(neither) = 1/3 ≈ 0.33333333
4. Check
- P(A or B) + P(neither) = 1 = 1.

Transcript

The problem. Here is the problem: Independent events have P(A) equals 0.5 and P(B) equals 1 over 3. Find P(A and B) and P(A or B). Two independent events A and B.

Step by step. Both: multiply, because the events are independent. At least one: add and subtract the overlap. Exactly one of them. Neither.

The answer. So the answer is P(A and B) equals 1 over 6 approximately 0.16666667; P(A or B) equals 2 over 3 approximately 0.66666667; P(neither) equals 1 over 3 approximately 0.33333333. For independent events the probability that both happen is the product of the individual probabilities.

Check. P(A or B) plus P(neither) equals 1 equals 1.

About probability

Probability of single events, complements, independent events and binomial outcomes.

Independent events have P(A) = 0.5 and P(B) = 1/3. Find…