The Basics of Probability Study Guide (page 2)
Introduction to The Basics of Probability
The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.
—CHARLES CALEB COLTON, English writer (1780–1832)
This lesson explores probability, presenting real-life examples and the mathematics behind them.
You've probably heard statements like, "The chances that I'll win that car are one in a million," spoken by people who doubt their luck. The phrase "one in a million" is a way of stating the probability, the likelihood, that an event will occur. Although most of us have used exaggerated estimates like this before, this chapter will teach you how to calculate probability accurately. Finding answers to questions like "What is the probability that I will draw an ace in a game of poker?" or "How likely is it that my name will be drawn as the winner of that vacation for two?" will help you decide if the probability is favorable enough for you to take a chance.
Probability is expressed as a ratio:
Example: When you toss a coin, there are two possible outcomes: heads or tails. The probability of tossing heads is therefore 1 out of the 2 possible outcomes:
Similarly, the probability of tossing tails is also 1 out of the 2 possible outcomes. This probability can be expressed as a fraction, , or as a decimal, 0.5. Since tossing heads and tails have the same probability, the two events are equally likely.
The probability that an event will occur is always a value between 0 and 1:
- If an event is a sure thing, then its probability is 1.
- If an event cannot occur under any circumstances, then its probability is 0.
For example, the probability of picking a black marble from a bag containing only black marbles is 1, while the probability of picking a white marble from that same bag is 0.
An event that is rather unlikely to occur has a probability close to zero; the less likely it is to occur, the closer its probability is to zero. Conversely, an event that's quite likely to occur has a probability close to 1; the more likely an event is to occur, the closer its probability is to 1.
Example: Suppose you put 2 red buttons and 3 blue buttons into a box and then pick one button without looking. Calculate the probability of picking a red button and the probability of picking a blue button.
The probability of picking a red button is or 0.4. There are 2 favorable outcomes (picking one of the 2 red buttons) and 5 possible outcomes (picking any of the 5 buttons). Similarly, the probability of picking a blue button is , or 0.6. There are 3 favorable outcomes (picking one of the 3 blue buttons) and 5 possible outcomes. Picking a blue button is more likely than picking a red button because there are more blue buttons than red ones: of the buttons are blue, while only of them are red.
Sometimes you might see probability written as a percentage: There is a 60% chance of thunderstorms tomorrow. What this means is or .
Probability with Several Outcomes
Consider an example involving several different outcomes.
Example: If a pair of dice is tossed, what is the probability of throwing a sum of 3?
- Make a table showing all the possible outcomes (sums) of tossing the dice:
- Determine the number of favorable outcomes by counting the number of times the sum of 3 appears in the table: 2 times.
- Determine the total number of possible outcomes by counting the number of entries in the table: 36.
Substitute 2 favorable outcomes and 36 total possible outcomes into the probability formula, and then reduce:
Throwing a 3 doesn't appear to be very likely with its probability of . Is any sum less likely than 3?
Probabilities That Add Up to 1
Think again about the example of 2 red buttons and 3 blue buttons from the previous section, Finding Probability. The probability of picking a red button is and the probability of picking a blue button is . The sum of these probabilities is 1.
The sum of the probabilities of every possible outcome of an event is 1.
Notice that picking a blue button is equivalent to NOT picking a red button:
Thus, the probability of picking a red button plus the probability of NOT picking a red button is1:
P(Event will occur) + P(Event will NOT occur) = 1
TipWhen there are only two outcomes possible, A or B, the probability that outcome A will occur is the same as 1 – P(Event B).
Example: A bag contains green chips, purple chips, and yellow chips. The probability of picking a green chip is and the probability of picking a purple chip is .What is the probability of picking a yellow chip? If there are 36 chips in the bag, how many are yellow?
|1. The sum of all the probabilities is 1:||P(green) + P(purple) + P(yellow) = 1|
|2. Substitute the known probabilities:|
|3. Solve for yellow:|
|The probability of picking a yellow chip is .|
|4. Thus, of the 36 chips are yellow:|
Thus, there are 15 yellow chips.
Gather the following coins together and put them into a box: 5 pennies, 3 nickels, 2 dimes, and 1 quarter. Without looking into the box, reach in to pull out an item. Before you touch any of the objects, figure out the probability of pulling out each item on your first reach.
The Basics of Probability Sample Questions
- What is the probability of throwing a sum of at least 7
- What is the probability of throwing a sum of 7 or 11?
Solutions to Sample Questions
- Determine the number of favorable outcomes by counting the number of table entries containing a sum of at least 7:
Sum # Entries 7 6 8 5 9 4 10 3 11 2 12
- Determine the number of total possible outcomes by counting the number of entries in the table: 36.
Substitute 21 favorable outcomes and 36 total possible outcomes into the probability formula:
Since the probability exceeds , it's more likely to throw a sum of at least 7 than it is to throw a lower sum.
There are two ways to solve this problem.
- Determine the number of favorable outcomes by counting the number of entries that are either 7 or 11:
Sum # Entries 7 6 11
- You already know that the number of total possible outcomes is 36. Substituting 8 favorable outcomes and 36 total possible outcomes into the probability formula yields a probability of for throwing a 7 or 11:
- Determine two separate probabilities—P(7) and P(11)—and add them together:
Since P(7 or 11) = P(7) + P(11), we draw the following conclusion about events that don't depend on each other:
P(Event A or Event B) = P(Event A) + P(Event B)
Find practice problems and solutions for these concepts at The Basics of Probability Practice Questions.
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