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X is binomial with n = 6 and p = 0.5. Find P(X = 3): animated lesson

A narrated, step-by-step probability video lesson with captions. Final answer: P(X = 3) ≈ 0.3125 (31.25%).

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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 (0.3 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Worked example
X is binomial with n = 6 and p = 0.5. Find P(X = 3).
2. Step by step
1. $P(X = j) = \binom{6}{j} (\frac{1}{2})^{j} (\frac{1}{2})^{6 - j}$
  Binomial probability of exactly j successes
2. $\binom{6}{3} = 20,\quad P(X = 3) \approx 0.3125$
  Substitute j = 3
3. $\mu = np = 3,\quad \sigma = \sqrt{np(1-p)} \approx 1.2247449$
  Mean and standard deviation of the distribution
3. The answer
$P(X = 3) \approx 0.3125$
P(X = 3) ≈ 0.3125 (31.25%)
4. Check
- All binomial probabilities for j = 0..6 add to 1.

Transcript

The problem. Here is the problem: X is binomial with n equals 6 and p equals 0.5. Find P(X equals 3). Binomial: k successes in n trials.

Step by step. Binomial probability of exactly j successes. Substitute j equals 3. Mean and standard deviation of the distribution.

The answer. So the answer is P(X equals 3) approximately 0.3125 (31.25 percent). With 6 independent trials and success probability 1 over 2 per trial, the requested probability is about 31.25 percent.

Check. All binomial probabilities for j equals 0..6 add to 1.

About probability

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

X is binomial with n = 6 and p = 0.5. Find P(X = 3) (Lesson)