Binomial Distributions
Binomial Probability distribution has the following characteristics:
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A fixed number of trials n. e.g., 20 coin toss , 40 students
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A binary outcome, a success and a failure. Probability of success is p, probability of failure is 1-p. e.g., head or tail in a coin toss , pass or fail of sttudents in an exam, positive and negative results of a test of a patient.
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Constant probability for each trial.e.g., The probability of a head in each trial is p, the probability of a pass or fail of a student is p1.
Coin Toss Example
5 coin tosses.
What is the probability that there are 3 Heads and 2 Tails ?
This is an example of a Binomial Probability Distribution.
Probability of a head is 0.5. One way to get 3 Heads and 2 Tails is HHHTTT. Probability is ${(0.5)}^{5}$. The other ways of having 3 Heads and 2 Tails is provided below.
| # | Outcome | Probability |
|---|---|---|
| 1 | HHHTT | ${(0.5)}^{5}$ |
| 2 | HHTHT | ${(0.5)}^{5}$ |
| 3 | HHTTH | ${(0.5)}^{5}$ |
| 4 | HHHTT | ${(0.5)}^{5}$ |
| 5 | THHTH | ${(0.5)}^{5}$ |
| 6 | THHHT | ${(0.5)}^{5}$ |
| 7 | HHHTT | ${(0.5)}^{5}$ |
| 8 | TTHHH | ${(0.5)}^{5}$ |
| 9 | THHTH | ${(0.5)}^{5}$ |
| 10 | THTHH | ${(0.5)}^{5}$ |
There are therefore ${5 \choose 3}$ ways of getting 3 heads and 2 tails. Therefore the probability is
${5 \choose 3}$ * ${(0.5)}^{3} * {(0.5)}^{2}$.
Therefore the probability using n trials and getting X success is ${n \choose X}$ * ${(p)}^{X} * {(1-p)}^{n-X}$.