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X is binomial with n = 8 and p = 0.25. Find P(X ≤ 1): animated lesson

A narrated, step-by-step probability video lesson with captions. Final answer: P(X ≤ 1) ≈ 0.3670806885 (36.708069%).

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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
X is binomial with n = 8 and p = 0.25. Find P(X ≤ 1).
2. Step by step
1. $P(X = j) = \binom{8}{j} (\frac{1}{4})^{j} (\frac{3}{4})^{8 - j}$
  Binomial probability of exactly j successes
2. $P(X \le 1) = \sum_{j=0}^{1} P(X = j) \approx 0.3670806885$
  Add the terms for j = 0 to 1
3. $\mu = np = 2,\quad \sigma = \sqrt{np(1-p)} \approx 1.2247449$
  Mean and standard deviation of the distribution
3. The answer
$P(X \le 1) \approx 0.3670806885$
P(X ≤ 1) ≈ 0.3670806885 (36.708069%)
4. Check
- All binomial probabilities for j = 0..8 add to 1.

Transcript

The problem. Here is the problem: X is binomial with n equals 8 and p equals 0.25. Find P(X less than or equal to 1). Binomial: k successes in n trials.

Step by step. Binomial probability of exactly j successes. Add the terms for j equals 0 to 1. Mean and standard deviation of the distribution.

The answer. So the answer is P(X less than or equal to 1) approximately 0.3670806885 (36.708069 percent). With 8 independent trials and success probability 1 over 4 per trial, the requested probability is about 36.7081 percent.

Check. All binomial probabilities for j equals 0..8 add to 1.

About probability

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

X is binomial with n = 8 and p = 0.25. Find P(X ≤ 1)