Probability I

The probability of an outcome of a random event is the proportion of times the outcome occurs, on average. Another way to think about probability is that, by repeating an event more and more times, the proportion of times the outcome occurs is a better and better approximation of the event. For example, to determine the probability of getting a heads in flipping a coin, the more times you flip a coin the more likely the proportion of heads you get is close to the actual probability. The probability of an outcome is a number between 0 and 1. A probability of 1 says it is 100% likely that the outcome will occur; in other words, the outcome always occurs. A probability of 0 says the outcome never occurs. A probability of .5 means that, on average, half of the time the outcome occurs. One of the complications of probability is that, by performing a random event several times, the actual outcome can be quite different than what would be predicted by knowing the probability of the outcome.

In this assignment we will investigate the probability of a few events, in order to get a feel for the concept.

Problem 1. Flip a coin 20 times and record how many heads and tails you get. What percentage of flips did you get a heads? Record the number of heads and the percentage of heads on the blackboard. What percentage of flips of the entire class were heads? Based on this, what would you estimate the probability of getting a heads on a flip to be? What do you think is the actual probability of getting a heads?

Problem 2. Roll a die 30 times and record how many occurrences of each number you got. Determine the percentage of rolls each number shows up. Record the number of occurrences of each number on the blackboard. What percentage of rolls of the entire class were of each number? Based on this data, what would you estimate the probability of rolling 1? What do you think is the actual probability?

In many situations one is interested in finding probabilities in a situation where there are several equally likely outcomes, such as flipping a coin or rolling a die. When all outcomes are equally likely, the probability of something occurring is given by

probability = (number of ways the outcome can occur) / (total number of outcomes)

For example, in rolling a die, there are 6 possible outcomes of what number is obtained. The probability of rolling a 3 is then 1/6; of the 6 possible outcomes, one of them is getting a 3. Similarly, the probability of getting a 4 or a 5 is 2/6 since there are 2 ways to get either a 4 or 5, and there are 6 total outcomes.

Problem 3. Suppose we are flipping a coin twice and recording the outcome. Write down all possible outcomes, listing them as first flip followed by second flip (e.g., heads, tails means a heads on the first flip and tails on the second flip). How many possible outcomes are there? Determine the probabilities of getting (a) 0 heads, (b) 1 heads, or (c) 2 heads in two flips.

Problem 4. Take one white die and one colored die, and write out all possible outcomes of rolling the two dice, listing each outcome with the white die's value first. How many outcomes are there? What are the possible sums of the two dice's values? Fill in the following table:

Sum	Probability
2
3
4
5
6
7
8
9
10
11
12

Which sums are the most likely? Which are the least likely?