Empirical Probability

Questions on Probability – Business Maths

Types of Probability

Probability can be seen in two ways. Empirical Probability and Theoretical Probability.

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  1. Empirical Probability forms the basis of inferential statistics. It has its origin in games of chance such as dice and card games. Many card players know from experience that certain types of hands turn up more frequently than the others. Relative frequency can be used to define empirical probability.
  2. Theoretical Probability. In practice, we do not derive all our values for probabilities from an experiment. Suppose we toss a die and we want to know the probability of throwing a 2. There are six ways the die can fall, of which only one is a 2. (kindly use the comment box below to ask a question).

Questions on Empirical & Theoretical Probability

  1. For each of the following pairs of events A and B say whether or not they are:
  • Mutually exclusive.
  • Exhaustive.
  • A child is chosen at random from a class.

A = {child has blue eyes}

B = {child has brown eyes)

  • A die is thrown.

A = {result is a multiple of 3}

B = {result is a multiple of 2}

  • A card is drawn from a pack of cards.

A = {the card is a picture card}

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B = {card is a king}

  • A coin is tossed

A = {toss gives a head}

B = {toss gives a tail}


  1. C and D are events such that:

P(C) = 0.7, P(D) = 0.6 and P(CD) = 0.8


  • P(CD)
  • P(C’D)
  • P(C or D but not both occurs)
  • P(C’D’)


  1. If A and B are two events which are exhaustive such that

P(AB) = ¼ , P(A|B) = 1/3


  • P(B)
  • P(A)
  • P(B|A)


  1. When three marksmen take part in a shooting contest, their chances of hitting a target are ½, 1/3, and ¼. Calculate the chance that one and only one bullet will hit the target if all men fire simultaneously.


  1. Two teams Legion Football Club Coker of Lagos (LFC) and Pam Pam Football Club Adeniji in Lagos State (PFC) play a football match against each other.

The probabilities for each team of scoring 0, 1, 2, 3 goals are shown in the table below:

(Kindly rotate your phone to view the table below)

0.3 0.2
1 0.3 0.4
2 0.3 0.3
3 0.1 0.1

Calculate the probability of:

  • Legion Football Club winning.
  • A draw.
  • Pam Pam Football Club winning.


  1. A factory has three machines 1, 2, and 3 producing a particular type of item. One item is drawn at random from the factory’s production. Let B denote the event that the chosen item is defective and let AK denote the event that the item was produced on machine k where k = 1, 2, or 3. Suppose that machines 1, 2, and 3 produce respectively 35%, 45%, and 20% of the total production of items and that…
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P(B|A1) = 0.02

P(B|A2) = 0.01

P(B|A3) = 0.03

Given that an item chosen at random is defective, find which machine was the most likely to have produced it.


  1. CharllyCARES Nigeria Ltd. Has a Quality Control department that examines finished output for faults. Any faulty has an 80% probability of being detected by the inspector, independent of whether any other fault is detected on either the same unit of output or different units.
  • One item of output contains three different faults, A, B, and C. What is the probability that:
  • All three faults will be detected?
  • No-fault will be detected.
  • Two faults will be detected, but the third will escape undetected?
  • The Factory manager now decides that the second stage of inspection should be added, in order to reduce the number of faulty units leaving the factory. If there is a 0.7 probability that faults remaining at this stage will be detected:
  • What is the probability that a unit of output with one fault will pass both stages of the inspection without the fault being found?
  • What is the probability that a unit of output with two faults will have one fault only detected at either stage of inspection, with the second fault escaping detection?
  • Of the total faults detected, what proportion will be detected at the stage of inspection and what proportion will be detected at the second stage?
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You may assume that every product is inspected at both stages. Even if a fault is found in the first stage.

Other Mathematics Questions

  1. Co-ordinate Geometry
  2. Linear Programming
  3. Permutation and Combination
  4. Set Theory
  5. Integration & Vector Algebra
  6. Differentiation

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