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ProbabilityStatistics and probabilityHigh schoolabout 0:51

Independent events have P(A) = 0.4 and P(B) = 0.25. 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) = 1/10 ≈ 0.1; P(A or B) = 11/20 ≈ 0.55; P(neither) = 9/20 ≈ 0.45.

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. 1. The problem
    Worked example
    Independent events have P(A) = 0.4 and P(B) = 0.25. Find P(A and B) and P(A or B).
  2. 2. Step by step
    1. P(A∩B)=P(A)P(B)=(25)(14)=110≈0.1P(A \cap B) = P(A)P(B) = (\frac{2}{5})(\frac{1}{4}) = \frac{1}{10} \approx 0.1
      Both: multiply, because the events are independent
    2. P(A∪B)=25+14−110=1120≈0.55P(A \cup B) = \frac{2}{5} + \frac{1}{4} - \frac{1}{10} = \frac{11}{20} \approx 0.55
      At least one: add and subtract the overlap
    3. P(A∪B)−P(A∩B)=920≈0.45P(A \cup B) - P(A \cap B) = \frac{9}{20} \approx 0.45
      Exactly one of them
    4. (1−P(A))(1−P(B))=920≈0.45(1 - P(A))(1 - P(B)) = \frac{9}{20} \approx 0.45
      Neither
  3. 3. The answer
    P(A∩B)=110≈0.1, P(A∪B)=1120≈0.55P(A\cap B) = \frac{1}{10} \approx 0.1,\ P(A \cup B) = \frac{11}{20} \approx 0.55

    P(A and B) = 1/10 ≈ 0.1; P(A or B) = 11/20 ≈ 0.55; P(neither) = 9/20 ≈ 0.45

  4. 4. Check
    • P(A or B) + P(neither) = 1 = 1.

Transcript

The problem. Here is the problem: Independent events have P(A) equals 0.4 and P(B) equals 0.25. 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 10 approximately 0.1; P(A or B) equals 11 over 20 approximately 0.55; P(neither) equals 9 over 20 approximately 0.45. 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.

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