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Understanding Probability: Concepts and Examples

Probability is a numerical measure representing the likelihood of an event occurring, expressed as a number between 0 and 1. Events are sets of outcomes from experiments, and the probability of an event happening is calculated by the number of favorable outcomes divided by the total possible outcomes. There are different types of events like mutually exclusive, non-mutually exclusive, independent, and dependent events. Probability laws include the Addition Law (OR Rule) and Multiplication Law (AND rule) for calculating probabilities of different types of events.

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Understanding Probability: Concepts and Examples

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  1. Numeracy and Data Analysis Probability

  2. What is Probability? Probability is a numerical measure of the likelihood that an event will occur. It is an idea which gives some notion, but may not be necessarily a definite one. Different terms can be used to convey the same idea as probability are chances, likelihood, odds, risk or hazard, random, certainty etc.

  3. Examples of use of Probability  High temperature is more likely in summer than winter. A healthy person has more chance of surviving from flue than an unhealthy person. Snow is likelihood in mountain areas in summer.

  4. What is an event? In probability, an event is a set of outcomes which generated as a result of an experiment. Examples of an event – Getting a head when tossing a coin Having an even number when rolling a dice

  5. What is an event? (Cont...) Probability of an event occurring can be presented as below. Example - If getting head when tossing can be presented as H, then probability of having head is P(H). If having an even number when rolling is E, then probability of getting an even number can be presented as P(E).

  6. Measurement of Probability In general, if an event E can occur in h ways out of total n ways, probability of having E can be presented as; P(E) = Number of possible ways E can be occurred Total number of possible outcomes P(E) = h n

  7. Measurement of Probability (Cont...)  Probability is measured on a scale of 0 to 1. 0 ≤ P(H) ≤ 1 where P(H) is the probability of an event H occurring.  Any impossible event has a probability 0 and any event which is definite has probability of 1. Note - It means probability of any event cannot be exceeded 1.

  8. Measurement of Probability (Cont...) Example 1: Calculate the probability of having ‘head’ when tossing a coin. A coin has two sides which are Head and Tail. It means total number of possible outcomes are 2. Probability of having Head which is; P(H) = 50% = ½ = 0.5

  9. Measurement of Probability (Cont...) Example 2: Calculate the probability of having number 5 when throwing a dice which has been numbered from 1 to 6. A dice has 6 sides which are 1, 2, 3, 4, 5 & 6. It means total number of possible outcomes are 6. Probability of having number 5 is; P(5) = 1/6 = 0.1667

  10. Measurement of Probability (Cont...) Example 3: A bag contains 100 of balls out of which 75 are red and 25 are blue. a) What is the probability of having a red ball? b) What is the probability of having a blue ball? a) Number of red balls are 75 Total number of balls are 100 Hence, Probability of having a red ball is; P(R) = 75/100 = 0.75 b) Number of blue balls are 25 Total number of balls are 100 Hence, Probability of having a blue ball is; P(B) = 25/100 = 0.25

  11. Types of Event 1) Mutually Exclusive Events – This means that we cannot get all the possible outcomes at once. It means either one or another only could happen. This can be expressed as; P(A and B) = 0 OR P(A ∩ B) = 0 Examples – When rolling a dice, getting a number , tossing a coin and having head and tail are mutually exclusive events.

  12. Types of Event (Cont...) 2) Non-Mutually Exclusive Events – This means that events can occur together. Example– When playing cards, having hearts and queens together is a non-mutually exclusive event. Because there is an one card having both hearts and queen.

  13. Types of Event (Cont...) 3) Independent Events – These events are not affected by another event. This can be expressed as; P(A) = P(A B) Example– Tossing a coin and having head or tail does not affect the next toss and having head or tail. In simple, the past toss does not affect the current toss.

  14. Types of Event (Cont...) 4) Dependent Events – These events can be affected by another event. Example– To be able to drive to work, car must be started. Driving to work depends on the ability of start the car engines on.

  15. The Two Laws of Probability There are TWO different probability laws as; 1) Addition Law (OR Rule) - For Mutually Exclusive Events - For Non-Mutually Exclusive Events 2) Multiplication Law (AND rule) - For Independent Events - For Dependent Events

  16. Addition Law (OR Rule): Mutually Exclusive Events In here, individual probabilities of mutually exclusive events are added together. This can be presented as; P (A or B) = P(A ᴜ B) = P(A) + P(B) Or can be presented using ᴜ symbol which is called as ‘union’.

  17. Addition Law (OR Rule): Mutually Exclusive Events (Cont...) Example 1 – What is the probability of having 2 OR 4 OR 5 when throwing a dice. Probability of having 2; P(2) = 1/6 Probability of having 4; P(4) = 1/6 Probability of having 5; P(5) = 1/6 Therefore, Probability of having 2 or 4 or 6; P(2 or 4 or 6) = 1/6 + 1/6 + 1/6 P(2 ᴜ 4 ᴜ 6) = 3/6 = 1/2 = 0.5

  18. Addition Law (OR Rule): Mutually Exclusive Events (Cont...) Example 2 – There are 100 balls in a bag in which 20 are blue, 30 are red and rest of them are green. What is the probability of having blue or green? Probability of having Blue; P(B) = 20/100 = 0.2 Probability of having Green; P(G) = 50/100 = 0.5 Therefore, Probability of having Blue OR Green; P(B or G) = 0.2 + 0.5 = 0.7

  19. Addition Law (OR Rule): Non-Mutually Exclusive Events Example 1 – In a supermarket, 70% of customers buy milk chocolates and 60% of them buy dark chocolates and 50% of them buy both types of chocolates. What proportion of customers buy at least one type of chocolate? Probability of buying milk chocolate; P(M) = 70/100 = 0.7 Probability of buying dark chocolate; P(D) = 60/100 = 0.6 Probability of buying both types; P(B) = 50/100 = 0.5 Therefore, Probability of buying at least one type of chocolate; P(M) + P(D) – P(B) 0.7 + 0.6 – 0.5 = 0.8

  20. Addition Law (OR Rule): Non-Mutually Exclusive Events (Cont...) Example 2 – A student is travelling 80% by train, 40% by bus and 30% by both streams. What is the probability of using at least one transport method. Probability by train; P(T) = 80/100 = 0.8 Probability by bus; P(B) = 40/100 = 0.4 Probability by both P (B and T) = 30/100 = 0.3 Therefore, probability of using at least one method; P(T) + P(B) – P(T and B) 0.8 + 0.4 – 0.3 0.9

  21. Multiplication Law (AND Rule): Independent Events In independent together. This can be presented as; here, individual events probabilities are of multiplied P (A and B) = P(A ∩ B) = P(A) × P(B) And can be presented using ∩ symbol which is called as ‘intersection’.

  22. Multiplication Law (AND Rule): Independent Events (Cont...) Example 1 – Tossing a coin in three different time period. What is the probability of having ‘tail’ on all three occasions? Probability of having tail at 1stattempt; P(T1) = ½ = 0.5 Probability of having tail at 2ndattempt; P(T2) = ½ = 0.5 Probability of having tail at 3rdattempt; P(T3) = ½ = 0.5 Therefore, probability of having tail at all three attempts; P (T1 and T2 and T3) = P(T1) × P(T2) × P(T3) P (T1 ∩ T2 ∩ T3) = 0.5 × 0.5 × 0.5 = 0.125

  23. Multiplication Law (AND Rule): Independent Events (Cont...) Example 2 – In textile manufacturing company, machine A produces 5000 items in which 500 are not in good quality. Machine B produces 8000 items in which 600 are not in good quality. If one item is taken from each machine, what is the probability that both items are not in good quality? Probability of having under quality items in machine A; P(A) = 500/5000 = 0.1 Probability of having under quality items in machine B; P(B) = 600/8000= 0.075 Therefore, probability of having under quality items in machine A and B; P (A and B) = P(A) × P(B) P(A ∩ B) = 0.1 × 0.075 = 0.0075

  24. Multiplication Law (AND Rule): Independent Events (Cont...) Example 3 – A die is thrown four different times. a) What is the probability of having number 6 in all four times? b) What is the probability of NOT having 6 in all four times? Probability of having six in first time; P(6first) Probability of having six in second time; P(6second) Probability of having six in third time; P(6third) Probability of having six in fourth time; P(6fourth) = 1/6 = 1/6 = 1/6 = 1/6 Answer a) Therefore, probability of having six in all four times; P (6firstand 6secondand 6thirdand 6fourth) P (6first∩ 6second∩ 6third∩ 6fourth) = P(6first) × P(6second) × P(6third) × P(6fourth) = 1/6 × 1/6 × 1/6 × 1/6 = 1/1296 = 0.00077 Probability of having six in all times Answer b) Probability of Not having 6 in all four times = 1 – 0.00077 = 0.99923

  25. Multiplication Law (AND Rule): Dependent Events Example 1 – In a bag of 100 balls, 20 are blue coloured. Two balls are to be selected one after another without replacement. What is the probability of selecting one blue ball followed by another blue ball? Probability that the first ball selected is blue; P(B1) = 20/100 = 0.2 Probability that the second ball selected is blue; P(B2) = 19/99 = 0.192 Therefore, probability of selecting one blue ball followed by another blue ball; P (B1 and B2) = P(B1) × P(B2) P (B1 ∩ B2) = 0.2 × 0.192 = 0.00384

  26. This is time for your questions... -THANK YOU-

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