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Calculating Binomial Probabilities (Edexcel International A Level Maths: Statistics 2)
Revision Note
Calculating Binomial Probabilities
Throughout this section we will use the random variable . For binomial, the probability of a X taking a non-integer or negative value is always zero. Therefore any values mentioned in this section will be assumed to be non-negative integers.
What are the tables for the binomial cumulative distribution function?
- In your formulae booklet you get tables which list the values of for different values of x, p and n
- can be 5, 6, 7, 8, 9 10, 12, 15, 20, 25, 30, 40, 50
- can be 05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5
- can be different values depending on n
- The probabilities are rounded to 4 decimal places
- The values of only go up to 0.5
- You can instead count the number of failures if the probability of success is bigger than 0.5
- Remember , which leads to identities:
How do I calculate, P(X = x) the probability of a single value for a binomial distribution?
- You can use the formula given in the formulae booklet
-
- The number of times this can happen is calculated by the binomial coefficient
-
- You can also use the tables for the Binomial Cumulative Distribution Function in the formulae booklet
How do I calculate, P(X ≤ x), the cumulative probabilities for a binomial distribution?
- If x is small, you could find the probability of each possible value of x and then add them together
- Otherwise, you will have to use the tables for the Binomial Cumulative Distribution Function in the formulae booklet
- If p is bigger than 0.5 then you will have to use the number of failures
How do I find P(X ≥ x)?
- : This means all values of X which are at least x
- These are all values of X except the ones that are less than x
- As x is an integer then as the probability of X is zero for non-integer values for a binomial distribution
- Therefore, to calculate :
- For example:
How do I find P(a ≤ X ≤ b)?
- : This means all values of X which are at least a and at most b
- This is all the values of X which are no greater than b except the ones which are less than a
- As X is an integer then as the for non-integer value of x for a binomial distribution
- Therefore to calculate :
- For example:
What if an inequality does not have the equals sign (strict inequality)?
- For a binomial distribution (as it is discrete) you could rewrite all strict inequalities (< and >) as weak inequalities (≤ and ≥) by using the identities for a binomial distribution
- and
- For example: and
- Though it helps to understand how they work
- It helps to think about the range of integers you want
- Always find the biggest integer that you want to include and the biggest integer that you then want to exclude
- For example, :
- You want the integers 5 to 10
- You want the integers up to 10 excluding the integers up to 4
- For example, P(X > 6) :
- You want the all the integers from 7 onwards
- You want to include all integers excluding the integers up to 6
- 1- P(X ≤ 6)
- For example, P(X < 8) :
- You want the integers 0 to 7
- P(X ≤ 7)
Worked example
The random variable . Find:
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
Examiner Tip
- Some calculators will calculate probabilities for binomial distributions
- These are great for checking your answers once you have answered your question showing the appropriate method
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