Probability Analysis of Coin Flipping and Die Rolling Experiments, Study notes of Mathematics

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Probability, continued

Data from Tuesday

Everybody flipped a coin 20 times and recorded the number of heads. Everybody rolled a die 20 times and recorded how many of each number they got.

Number of Students with a given number of Heads 0 2 4 6 8 10 12 14 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Number of Heads Number of Students

For the die rolling, the following table lists the possible outcomes, the total number of times the number was achieved, and the fraction of the total number of rolls die 1 2 3 4 5 6 number 127 161 139 144 131 141 fraction 0.15 0.19 0.16 0.17 0.16 0.

Flip two coins 20 times and record the number of times you got two heads, the number of times you got two tails, and the number of times you got one of each. Tabulate your data. Do you think the probability of getting two heads is the same as the probability of getting one of each?

Roll two dice 20 times and record the sum of the two on each roll. Tabulate your data on the sheet going around the class. Do you think the probability of rolling a 7 is the same as rolling a 12? Why or why not?

To get the probability of each outcome, you multiply the various probabilities along the routes to get to the outcome. In this case, we get .5 * .5 = .25 chance for each of the four outcomes. Getting one of each is then two of the four outcomes, and the combined probability is .25 + .25 = .50.

Equally Likely Events

The most common situation in probability is when all possible outcomes are equally likely. Examples are flipping a coin and rolling a die.

To compute the probability of something happening, when you have equally likely outcomes, you can use the following formula: Prob. = number of ways the outcome can occur total number of outcomes

Question

How many outcomes are there when you roll two dice?

This question does not have a unique answer; however, how you answer it affects how to calculate probabilities, and one way (possibly the most natural way), makes the calculation simpler. Think about having dice of two different colors, say red and green. Then the outcomes are in the following table:

There are then 36 total outcomes. One way to think about this rather than listing all of them, is that there are 6 possibilities for the green die, and 6 for the red die. What happens on the green die does not affect what can happen on the red die. In this case, we can get the total outcomes by multiplying: 6 * 6 = 36 total outcomes.

Each of the 36 outcomes are equally likely, so the probability of any one happening is 1/36, which is a little less than 3%.