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Probabilities give us an idea of how likely it is for a certain event to happen. For example, when a coin is flipped, the chance that it comes up heads is 50%. Probabilities can be expressed as decimals, fractions, percents, or ratios. We could have said the probability of flipping heads is , 0.5, 1/2, 50% or 1:2. Each of these conveys the idea that we should expect to get a heads half of the time. Probabilities only give us an idea of what to expect in the long run. However, they do not tell us what will happen in the short term.
[Figure2]
Suppose we flip a coin 10 times in a row and get heads each time. The next coin flip is still a random event because while we cannot tell for certain what the next flip will be, we can be certain that about 50% of all tosses over a long set of tosses will be heads. Some people think that we are on a roll so we are more likely to get another heads. Others will say that getting tails is more likely because we are due to get tails. The truth is that we cannot tell what will happen on the next flip. The only thing we know for certain is that there is a 50% chance that the coin will be heads on its next flip. If we continue to flip this same coin hundreds of times, we would expect the percent of heads to get closer and closer to 50%.
Chance Behavior is not predictable in the short term, however, it has long term predictability. The Law of Large Numbers tells us that despite the results on a small number of flips, we will eventually get closer to the theoretical probability. The outcomes in any random event will always get close to the theoretical probability if the event is repeated a large number of times. We might roll a die 4 times in a row and get a 6 each time, however, if we rolled this die hundreds of times, the percent of time that we get a 6 will get closer and closer to the theoretical probability of 1/6. When calculating a probability, we divide the number of favorable outcomes (outcomes we are interested in) by the total number of outcomes. In other words, the probability that outcome 'A' occurs is found by the formula
Consider a standard deck of 52 playing cards.
[Figure4]
If we asked the question "What is the probability of being dealt a face card (jack, queen, or king)?", we would need to count how many cards are face cards and then divide by the total number of cards, 52.
In this situation there are 12 face cards and 52 cards overall so our probability of getting a face card is 12/52 or 3/13. That is approximately 0.23.
In probability, there are outcomes that are sure to happen and there are outcomes that are impossible. If we are once again dealing with a standard 52 card deck, the chance of being dealt either a red card or a black card if one card is dealt is 100%. The chance of being dealt a blue card is 0% since there are no blue cards in a standard deck. All random events have probabilities between 0 and 1. In addition, the sum total of the probabilities for all possible outcomes in the sample space is equal to 1. In other words, if an event occurs, there is a 100% chance that one of the possible outcomes will happen. The list below summarizes these rules.
a) The probability of a sure thing is 1.
b) The probability of an impossible outcome is 0.
c) The sum of the probabilities of all possible outcomes is 1.
d) The probability for any random event must be somewhere from 0 to 1.
As shown earlier, we notate the probability of event 'A' happening as P(A). For example, the probability of rolling a three on a six-sided die can be written
Sometimes we are interested in the probability of an event not occurring. This is called the complement of the event. We can write the probability of the complement of event 'A' happening as P(~A), P(not A), or
The formula for the complement of an event is
On our die rolling question, P(~3). In other words, there is a 5/6 chance of the dice not landing on a 3. It is important to notice that the probability of an event happening and the probability of its complement always add up to 1.
Example 1
Which of the following situations are random events?
i) A student looks through their closet to decide what shirt to wear to school.
ii) A student labels each of their 6 pairs of shoes 1 through 6 and then rolls a single die to decide which pair to wear.
iii) The state legislature decides to increase funding to schools by 3%.
iv) A professional golfer makes a hole-in-one on a 200 yard hole.
Solution
Situations i) and iii) are not random events. In both cases, there are additional factors that are influencing the decision. The day of the week or the temperature outside might influence your shirt choice and how much money the state legislature happens to have might influence funding.
Both situations ii) and iv) are random events because while we can't predict what will happen in this particular instance, we can make long term predictions. We can predict the percent of the time the student might end up with the shoes labeled #2 and we can predict the percent of the time that the golfer will make a hole-in-one based upon previous performance.
Example 2
In the game of pool, there are a total of 15 balls. Balls numbered 1-8 are solid and balls 9-15 are striped. There are two pool balls of each color, for example, there are two yellow pool balls. One of those are solid and one of those are striped. The only exception to this is that there is only 1 black pool ball, the eight ball, and it is solid.
[Figure6]
Suppose the pool balls were put in a bag and a single pool ball is pulled out of the bag. What is the probability that the ball:
a) is yellow?
b) is striped?
c) has a number on it that is greater than 10?
d) is not striped?
Solution
a)
b)
c)
d) P(~Striped)=
In addition to these types of questions, we can also calculate probabilities by incorporating our counting methods from Unit 1. Recall that the probability of an event occurring is the number of favorable outcomes divided by the number of total possible outcomes.
Example 3
A jury of 12 people is to be selected from a group of 12 men and 8 women. What is the probability that the jury has at least 6 women on it?
Solution
The total number of outcomes possible is based upon selecting 12 members from a pool of 20. Since order will not matter, there are 20C12 = 125,970 ways to pick a jury of 12. We now want to have at least 6 women on the jury. This means we could have 6 women and 6 men or 7 women and 5 men or 8 women and 4 men on the jury. Mathematically, this would be
There are 32,703 ways to have at least 6 women on the jury out of a possible total of 125,970 different juries or 32,703/125,970. There is about a 26% chance that the jury will have at least 6 women on it.
Sometimes, data is organized in a Venn diagram, as shown in Example 4 on the following page. We will examine these in greater depth in section 2.3 but for now, it is important to understand that a Venn diagram is an organizational tool that makes it easier to interpret a situation and answer basic probability questions.
Example 4
A class of 30 students is surveyed to see whether or not they had a science class and/or a math class this trimester. There are 18 students that have a math class, 14 students who have a science class, and 4 students who have neither. It also turns out that this includes 6 students who currently have both classes. The results of the survey are shown in the Venn diagram below.
a) How many total students are taking a math class this trimester?
b) What is the probability that a randomly selected student is taking a math class this trimester?
c) What is the probability that a randomly selected student is taking both a math and science class this trimester?
d) What is the probability that a randomly selected student is not taking either a math or science class this trimester?
Solution
a) There are 12 kids who only have a math class and 6 kids who have both a math a science class this trimester for a total of 18 kids.
b)
c)
d)
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