Probability Terminology: Experiment (or trial/s): In probability theory, an experiment or trial is any procedure that can be repeated as often as desired, and has a well-defined set of results. Outcome: Result or results of an experiment. Event: A set or sets of outcomes. Sample Space: Set of all possible outcomes. Probability: The measure of how likely is an event.

Relative frequency (or empirical probability): The ratio of outcomes to the total number of trials. For example, toss a die 𝑛 times and record the number of times 𝑘 that you get a 2. The relative frequency is then 𝑘 𝑛 . This relative frequency approaches the actual probability as the number of trials increase indefinitely, that is, 𝐥𝐢𝐦 𝒏→∞ 𝒌 𝒏 =𝑷(𝟐) , where 𝑃(2) is the actual probability of obtaining a 2 given any unbiased toss of a die. What is 𝑃 2 ? The probability of an event 𝐸, say 𝑃(𝐸) is the sum of the probabilities of the outcomes which make up 𝐸. Complement of an event. If 𝐸 is an event, then 𝐸 ′ or not 𝑬 is the complement of 𝐸, where 𝑃 𝐸 +𝑃 𝐸 ′ =1 Mutually exclusive events have no common outcome. For example, it’s impossible to pass and fail an exam at the same time. If events are mutually exclusive, then their probabilities can be summed, that is, the probability of any of these events taking place is: 𝑃 𝐸 1 +𝑃 𝐸 2 +𝑃 𝐸 3 +…+𝑃 𝐸 𝑛

𝑃 𝐷 = 𝑛(𝐷) 𝑛(𝑆) = 3 6 = 1 2 Formal definition of probability: 𝑃 𝐸 = The number of ways event 𝐸 can occur Total number of possible outcomes Both the number of ways an event can occur and the total number of possible outcomes form sets. The number of ways event E can occur is the set of desirable outcomes. The total number of possible outcomes is the set of all possible outcomes. Example: If a fair die is thrown and we want to know the probability that an even number will occur, then the set of desirable outcomes is D={2, 4, 6}. The set of total possible outcomes is S={1, 2, 3, 4, 5, 6}. 𝑃 𝐷 = 𝑛(𝐷) 𝑛(𝑆) = 3 6 = 1 2 Thus the probability of obtaining an even number is 50%.

Now do Exercise 4A. 

A fair die is thrown once. Find the probability that the score is Bigger than 3 Bigger than or equal to 3 An odd number A prime number Bigger than 3 and a prime number Bigger than 3 or a prime number or both Bigger than 3 or a prime number, but not both 𝑃 𝐸 = The number of ways event 𝐸 can occur Total number of possible outcomes Soln: S E>3 ={4, 5, 6} T E ={1,2,3,4,5,6} P E = n(S) n(T) = 3 6 = 1 2 P E = 4 6 = 2 3 c. P E = 3 6 = 1 2 d. P E = 3 6 = 1 2 e. P E = 1 6 f. P E = 3 6 = 1 2 𝑆(𝐸)={4,5,6} g. P E = 2 6 = 1 3 𝑆(𝐸)={4,6}

2. A card is chosen at random from an ordinary pack 2. A card is chosen at random from an ordinary pack. Find the probability that the card is Red A picture (K, Q J) An honour (A, K, Q, J, 10) A red honour Red or honour or both Soln: P E = 26 52 = 1 2 b. P E = 12 52 = 3 13 c. P E = 20 52 = 5 13 d. P E = 10 52 = 5 26 P E = 36 52 = 9 13 n 𝑆 𝐸 =𝑛 𝑅𝑒𝑑 𝑐𝑎𝑟𝑑𝑠 + 𝑛(𝐵𝑙𝑎𝑐𝑘 ℎ𝑜𝑛𝑜𝑢𝑟 𝑐𝑎𝑟𝑑𝑠)

3. Two fair die are thrown simultaneously. Find the probability that The total is 7 The total is at least 8 The total is a prime number Neither of the scores is a 6 At least one of the scores is a 6 Exactly one of the scores is a 6 The two scores are the same The difference between the scores is an odd number 𝑇(𝐸) = { 1,1 , 1,2 , 1,3 , 1,4 , 1,5 , 1,6 , 2,1 , 2,2 , 2,3 , 2,4 , 2,5 , 2,6 , 3,1 , 3,2 , 3,3 , 3,4 , 3,5 , 3,6 , 4,1 , 4,2 , 4,3 , 4,4 , 4,5 , 4,6 , 5,1 , 5,2 , 5,3 , 5,4 , 5,5 , 5,6 , 6,1 , 6,2 , 6,3 , 6,4 , 6,5 , 6,6 } Soln: P E = 6 36 = 1 6 𝑆 𝐸 = 1,6 , 6,1 , 2,5 , 5,2 , 3,4 , 4,3 b. P E = 15 36 = 5 12 c. P E = 15 36 = 5 12 d. P E = 25 36 e. P E = 11 36 f. P E = 10 36 = 5 18 g. P E = 6 36 = 1 6 h. P E = 18 36 = 1 2

4. A fair die is thrown twice 4. A fair die is thrown twice. If the second score is the same as the first, the second throw does not count, and the die is thrown again until a different score is obtained. The two different scores are added to give a total. List the outcomes and find the probability that The total is 7 The total is at least 8 At least one of the two scores is a 6 The first score is higher than the last Soln: 𝑇(𝐸) = { 1,2 , 1,3 , 1,4 , 1,5 , 1,6 , 2,1 , 2,3 , 2,4 , 2,5 , 2,6 , 3,1 , 3,2 , 3,4 , 3,5 , 3,6 , 4,1 , 4,2 , 4,3 , 4,5 , 4,6 , 5,1 , 5,2 , 5,3 , 5,4 , 5,6 , 6,1 , 6,2 , 6,3 , 6,4 , 6,5 } a. P E = 6 30 = 1 5 b. P E = 12 30 = 2 5 c. P E = 10 30 = 1 3 d. P E = 15 30 = 1 2

5. A bag contains 10 counters, of which 6 are red and 4 are green 5. A bag contains 10 counters, of which 6 are red and 4 are green. A counter is chosen at random; its colour is noted and it is replaced in the bag. A second counter is then chosen at random. Find the probability that Both counters are red Both counters are green Just one counter is red At least one counter is red The second counter is red Soln: P E = 36 100 = 9 25 b. P E = 16 100 = 4 25 c. P E = 48 100 = 12 25 d. P E =1− 4 25 = 21 25 e. P E = 60 100 = 3 5

6. Draw a bar chart to illustrate the probabilities of the various total scores when two fair die are thrown simultaneously. Soln: