Explanation
Definition
The binomial distribution is a discrete probability distribution. It describes how often in n trials with a success probability of p a certain event occurs. The random variable X counts the successes.
P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}
Procedure
Scheme
Calculate the probability of a specific event using the binomial formula.
P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}
Calculate the expected value, which gives the mean of the distribution.
E(X) = n \cdot p
For n = 20 and p = 0{,}3: Calculate E(X) = 20 \cdot 0{,}3 = 6.
Calculate the standard deviation, which describes the spread of the values.
\sigma = \sqrt{n \cdot p \cdot (1-p)}
For n = 20 and p = 0{,}3: \sigma = \sqrt{20 \cdot 0{,}3 \cdot 0{,}7} \approx 2{,}05.
Missunderstandings
Common Errors
- ★It is often forgotten to calculate the combination \binom{n}{k} correctly.
- ★It is often overlooked that p and 1 - p together 1 result in.
- ★It is often a rounding error when calculating the standard deviation.
Problems
1 / 3
Calculate
P(X = 2) for n = 5,\, p = 0{,}4
2 / 3
Calculate the expected value for
n = 15,\, p = 0{,}5
3 / 3
Calculate the standard deviation for
n = 25,\, p = 0{,}2
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