Edexcel A Level Maths: Statistics:复习笔记4.2.2 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.

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

• You should have a calculator 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 'value - the value of  for which you want to find
  • The value - the number of trials
  • The 'value - the probability of success
• Some calculators will give you the option of listing the probabilities for multiple values  of at once
• There is a formula that you can use but you are expected to be able to use the distribution function on your calculator
  • 
    • If there are successes then there are  failures
    • The number of times this can happen is calculated by the binomial coefficient
    • This can be seen by considering a probability tree diagram with n trials, where p is the probability of success and the tree diagram is being used to find x successes
    •  is the number of pathways through the tree there would be exactly x successes within the n trials
  • You might find it quicker to use the formula than finding using the binomial probability distribution function on your calculator

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

• You should have a calculator that can calculate cumulative binomial probabilities
  • Most calculators will only find
  • Some calculators can find
• You want to use the "Binomial Cumulative Distribution" function
  • This is sometimes shortened to BCD, Binomial CD or Binomial Cdf
• You will need to enter:
  • The 'x' value - the value of x for which you want to find
    • Some will instead ask for lower and upper bounds
    • For this lower would be 0 and upper would be x
  • Thevalue - the number of trials
  • The value - the probability of success

How do I find P(X ≥ x)?

• You might be lucky enough to have a calculator that has lower and upper bounds:
  • Use  for the lower bound and  for the upper bound
  • Otherwise, you will need some extra identities
• : This means all values of X which are at least x
  • This is 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)?

• You might be lucky enough to have a calculator that has lower and upper bounds:
  • Use a for the lower bound and b for the upper bound
  • Otherwise, you will need some extra identities
• : 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 probability of X  is zero for non-integer values 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
• • 
  • 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)

How do I use the binomial cumulative distribution function tables?

• In your formula booklet you get tables which list the values of P(X ≤ x)for different values of x, p and n
  • n can be 5, 6, 7, 8, 9 10, 12, 15, 20, 25, 30, 40, 50
  • p can be 05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5
  • x can be different values depending on n
• The probabilities are rounded to 4 decimal places
  • If you want more accurate values then you will need to use your calculator
• The tables are useful when you want to find a value of x given the probability
  • For example, the largest value of such that P(X ≤ x) is less than 0.95
• You can estimate P(X = k  )using the tables by using:
  • 
  • To get a more accurate estimate use the formula or the binomial probability distribution function on your calculator
• The values of p 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 X+Y =n, which leads to identities:
    • 
    • 
    •