Edexcel A Level Maths: Statistics:复习笔记4.2.2 Calculating Binomial Probabilities

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:

Worked Example

The random variable . Find:

(a)
(b)
(c)
(d)

Exam Tip

  • Always make sure you are using the correct function on your calculator. Most questions will be in context so try and pick out the key words and numbers. If the question is worth more than one mark then be sure to show a method to get at least one mark if you write the answer down incorrectly.

 

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