Definition of a ProbabilitySuppose an event E can happen in r ways out of a total of n possible equally likely ways. Show
Then the probability of occurrence of the event (called its success) is denoted by
The probability of non-occurrence of the event (called its failure) is denoted by
Notice the bar above the E, indicating the event does not occur. Thus,
In words, this means that the sum of the probabilities in any experiment is `1`. Definition of Probability using Sample SpacesWhen an experiment is performed, we set up a sample space of all possible outcomes. In a sample of N equally likely outcomes we assign a chance (or weight) of `1/N` to each outcome. We define the probability of an event for such a sample as follows:
That is:
where
Properties of Probability(a) 0 ≤ P(event) ≤ 1
(b) P(impossible event) = 0
(c) P(certain event) = 1
Example 1What is the probability of... (a) Getting an ace if I choose a card at random from a standard pack of `52` playing cards. Answer In a standard pack of 52 playing cards, we have: ♥
2 3 4 5 6 7 8 9 10 J Q K A There are 4 aces in a normal pack. So the probability of getting an ace is:
(b) Getting a `5` if I roll a die. Answer A die has 6 numbers. There is only one 5 on a die, so the probability of getting a 5 is given by:
(c) Getting an even number if I roll a die. Answer Even numbers are `2, 4, 6`. So
(d) Having one Tuesday in this week? Answer Each week has a Tuesday, so probability = `1`. Example 2There are `15` balls numbered `1` to `15`, in a bag. If a person selects one at random, what is the probability that the number printed on the ball will be a prime number greater than `5`? Answer The primes between `5` and `15` are: `7, 11, 13`. So the probability `=3/15=1/5` Example 3The names of four directors of a company will be placed in a hat and a 2-member delegation will be selected at random to represent the company at an international meeting. Let A, B, C and D denote the directors of the company. What is the probability that (a) A is selected? (b) A or B is selected? (c) A is not selected? Answer The possible outcomes are: AB, AC, AD, BC, BD, CD. [There are a few explanations for each answer - hopefully at least one of them makes sense!] Part (a)Explanation 1: The probability is `3/6=1/2` since when we choose A, we must choose one of the remaining 3 directors to go with A. There are `C_2^4=6` possible combinations. Explanation 2: Probability that A is selected is `{C_1^1 times C_1^3}/{C_2^4} = 3/6 = 1/2` [Choose A (`C_1^1`), and then choose one from the 3 remaining directors (`C_1^3`), divided by the number of possible outcomes: `C_2^4`.] Part (b)Explanation 1: The probability of getting A or B first is `2/4=1/2`. Now to consider the probability of selecting A or B as the second director. In this case, the first director has to be C or D with probability `2/4` (2 particular directors out of 4 possible). Then the probability of the second being A or B is `2/3` (2 particular directors out of the remaining 3 directors). We need to multiply the two probabilities. So the probability of getting A or B for the second director is `2/4 xx 2/3 = 1/3` The total is: `1/2 + 1/3 = 5/6` Explanation 2: Probability that A or B is selected is `frac{C_1^1 times C_1^3 + C_1^1 times C_1^2}{C_2^4}` `=frac{3+2}{6}` `=5/6` [Choose A as above, then choose B from the remaining 2 directors in a similar way.] Explanation 3: If A or B is chosen, then we cannot have the case C and D is chosen. So the probability of A or B is given by: `P("A or B") = 1-P("C and D")` `=1-1/6` `=5/6` Part (c)Probability that A is not selected is `1-1/2=1/2` ExtensionConsider the case if we are choosing 2 directors from 5. The probabilities would now be: (a) Probability that A is selected is `frac{C_1^1 times C_1^4}{C_2^5}=4/10=2/5` [Choose A (`C_1^1`), and then choose one from the 3 remaining directors (`C_1^4`), divided by the number of possible outcomes: `C_2^5`.] (b) Probability that A or B is selected is `frac{C_1^1 times C_1^4 + C_1^1 times C_1^3}{C_2^5}` `=frac{4+3}{10}` `=7/10` [Choose A as above and then choose B from the remaining 3]. (c) Probability that A is not selected is `1-2/5=3/5`. Coming next...♥ 2 3 4 5 6 7 8 9 10 J Q K A The next 2 sections give more examples of probability:
How do you find the total number of outcomes for an event?To find the total number of outcomes for two or more events, multiply the number of outcomes for each event together. This is called the product rule for counting because it involves multiplying to find a product.
What is the total number of outcomes called?The set of all possible outcomes of an experiment is called the sample space. Events are subsets of the sample space, and they are assigned a probability that is a number between zero and one, inclusive.
Which of the following refers to the ratio of the number of ways the event can occur to the number of outcomes?Odds are used to describe the chance of an event occurring. The odds are the ratios that compare the number of ways the event can occur with the number of ways the event cannot occurr. The odds in favor - the ratio of the number of ways that an outcome can occur compared to how many ways it cannot occur.
What compares the number of events that occur to the total number of possible events?Probability is a type of ratio where we compare how many times an outcome can occur compared to all possible outcomes.
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Which refers to the number of ways an event can occur to the total number of possible outcomes?
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