PROBABILITY

If there are only 2 red marbles in a jar containing exactly 5 marbles, the probability of randomly picking a red marble out of the jar is 2 out of 5, or .

The probability of an event =

In this case, the probability of randomly choosing a red marble, or

P(Red) = =

If the rest of the marbles are blue, it must be true that the probability of choosing either a red or a blue marble is 100% or 1. Then the probability of choosing a blue marble

P(Blue) = , since P(Red) + P(Blue) = + = 1.

Example: If three coins are tossed, what is the probability of getting exactly two heads?

Answer:

Solution: Make a list. There are 8 possible outcomes. It helps to picture three different coins such as a penny, a dime, and a quarter, so that we can see that the outcome HTT is not the same as the outcome THT, for example.

HHH HHT HTH HTT

THH THT TTH TTT

Three of the outcomes have 2 tails and 1 head, so the probability of getting exactly two tails is .

Example: What is the probability of getting 3 heads?

Answer:

Solution: We can use the same list as above, but if the problem asked for, say, the probability of gettng 5 heads, we would want a shortcut.

P(first coin is a head) = since there are two possible outcomes for a coin toss.

P (first and second coin both heads) = . There is a chance that the first coin will be heads and half again as much of a chance of the second coin being heads also.

P (all 3 heads) =

Example: Three coins are tossed. If the first two coins are heads, what is the probability that the third coin is a head?

Answer:

Solution: It doesn't matter how the first two coins have landed when the third coin is tossed. The third coin will still have a 50/50 chance of being heads. Notice how this problem differs from one in which we are betting in advance that all three coins will be heads.

Note: A famous expression relating to probability is, "Dice have no memory." In this case we can say, "Coins have no memory." What this means is that regardless of previous tosses, any coin toss still has a 50% probability of being heads. Even if you toss 100 heads in a row, the probabilty of the next toss being heads is still 50%! In this problem, the other two coins being heads has no effect on the probability that the third coin is also heads.

Example: If a marble is randomly chosen from a jar with only red, green, and pink marbles, the probability that it will be red is and the probability that it will be green is .

Question 1: What is the probability that it will be pink?

Question 2: What is the fewest number of marbles that could be in the jar?

Answer 1:

Solution: If we randomly choose a marble from the jar there is a 100% probability that it will be some color. It must be either red, green, or pink. In other words, the sum of the probabilities of choosing either a red, green or pink marble is 1.

P(red) + P(green) + P(pink) = 1

Let x be the probability of choosing a pink marble.

Then + + x = 1

Þ + x = 1

Þ x =

Answer 2: 6

Solution:

P(Red) = =

P(Blue) = =

P(Pink) = =

Since in each case the total number of marbles is the same, we are looking for the least common multiple (or the lowest common denominator) of 2, 3, and 6, which is 6.

Check: If there are six marbles then 3 are red, 2 are blue, and 1 is pink.

P(Red) = = =

P(Blue) = = =

P(Pink) = = =