Calculating Binomial Probabilities (DP IB Analysis & Approaches (AA): HL): Revision Note

Calculating binomial probabilities

Throughout this section we will use the random variable $X~\mathrm{B}(n,p)$. For a binomial distribution, the probability of $X$ taking a non-integer or negative value is always zero. Therefore any values of $X$ mentioned in this section will be assumed to be non-negative integers: 0, 1, 2, 3, 4, etc.

How do I calculate P(X = x): the probability of a single value for a binomial distribution?

• You should have a GDC that can calculate binomial probabilities

• You want to use the "Binomial Probability Distribution" function

  • This is sometimes shortened to BPD, Binomial PD or Binomial Pdf

• You will need to enter:

  • The 'x' value - the value of x for which you want to find $\mathrm{P}(X=x)$

  • The 'n' value - the number of trials

  • The 'p' value - the probability of success

• Some calculators will give you the option of listing the probabilities for multiple values of x at once

• There is a formula that you can use but you are expected to be able to use the distribution function on your GDC

  • $\mathrm{P}(X=x)={}^n\mathrm{C}_x\times p^x{(1-p)}^{n-x}$

    • ${}^n\mathrm{C}_x=\frac{n!}{x!(n-x)!}$

How do I calculate P(a ≤ X ≤ b): the cumulative probabilities for a binomial distribution? 

• You should have a GDC that can calculate cumulative binomial probabilities

  • Most calculators will find $\mathrm{P}(a\le X\le b)$

  • Some calculators can only find $\mathrm{P}(X\le b)$

    • The identities below will help in this case

• You should use the "Binomial Cumulative Distribution" function

  • This is sometimes shortened to BCD, Binomial CD or Binomial Cdf

• You will need to enter:

  • The lower value - this is the value a

    • This can be zero in the case $\mathrm{P}(X\le b)$

  • The upper value - this is the value b

    • This can be n in the case $\mathrm{P}(X\ge a)$

  • The 'n' value - the number of trials

  • The 'p' value - the probability of success

How do I find probabilities if my GDC only calculates P(X ≤ x)?

• To calculate P(X ≤ x) just enter x into the cumulative distribution function

• To calculate P(X < x) use:

  • $\mathrm{P}(X<x)=\mathrm{P}(X\le x-1)$ which works when X is a binomial random variable

    • P(X < 5) = P(X ≤ 4)

• To calculate P(X > x) use:

  • $\mathrm{P}(X>x)=1-\mathrm{P}(X\le x)$ which works for any random variable X 

    • P(X > 5) = 1 - P(X ≤ 5)

• To calculate P(X ≥ x) use:

  • $\mathrm{P}(X\ge x)=1-\mathrm{P}(X\le x-1)$ which works when X is a binomial random variable

    • P(X ≥ 5) = 1 - P(X ≤ 4)

• To calculate P(a ≤ X ≤ b) use:

  • $\mathrm{P}(a\le X\le b)=\mathrm{P}(X\le b)-\mathrm{P}(X\le a-1)$ which works when X is a binomial random variable

    • P(5 ≤ X ≤ 9) = P(X ≤ 9) - P(X ≤ 4)

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:

  • $\mathrm{P}(X<x)=\mathrm{P}(X\le x-1)$ and $\mathrm{P}(X>x)=\mathrm{P}(X\ge x+1)$

  • For example: P(X < 5) = P(X ≤ 4) and P(X > 5) = P(X ≥ 6)

• It helps to think about the range of integers you want

  • Identify the smallest and biggest integers in the range

• If your range has no minimum or maximum then use 0 or n

  • $\mathrm{P}(X\le b)=\mathrm{P}(0\le X\le b)$

  • $\mathrm{P}(X\ge a)=\mathrm{P}(a\le X\le n)$

• $\mathrm{P}(a<X\le b)=\mathrm{P}(a+1\le X\le b)$

  • P(5 < X ≤ 9) = P(6 ≤ X ≤ 9)

• $\mathrm{P}(a\le X<b)=\mathrm{P}(a\le X\le b-1)$

  • P(5 ≤ X < 9) = P(5 ≤ X ≤ 8)

• $\mathrm{P}(a<X<b)=\mathrm{P}(a+1\le X\le b-1)$

  • P(5 < X < 9) = P(6 ≤ X ≤ 8)