Multiple Probabilities

The probability of 2 things happening:
The probability of 2 independent things happening either one after the other or together is the probability of the first thing happening multiplied by the probability of the second thing happening.

Example 1: Flipping a coin 2 times:
What is the probability of flipping a coin 2 times and it coming up "heads" both times?
            first       second
            toss:        toss:
              1           1      1
  Answer:    ---    X    ---  = --- or 1 chance in 4 tries
              2           2      4
Note: This answer of 1/4 is correct before you make the first toss. After you toss the coin once, and it comes up heads, the probability that the second toss will also come up heads is 1/2 because the first toss is in the past.

Example 2: Drawing 2 cards at the same time

What is the probability of drawing 2 aces in a row out of a deck of 52 cards, if you don't put the first card back in the deck before drawing the second card?

            first       second
            draw:        draw:
              4           3      12    1
  Answer:    ---    X    ---  = --- = --- or 1 chance in 221 tries
              52          51    2652  221
Why is the probability of the second ace 3/51?
Well, after you draw the first ace (you assume you got it) there are only 3 aces left and 51 cards.

Example 3: Drawing 2 cards with replacement:
How would the probabilities change if you had put the first ace back in the deck before drawing the second time? Well, this makes the probability of drawing an ace on the second draw the same as on the first draw, so the probability of drawing 2 aces with replacement of the first ace is:
            first       second
            draw:        draw:
              4           4      16    1
  Answer:    ---    X    ---  = --- = --- or 1 chance in 169 tries
              52          52    2704  169
Since 1/169 is larger than 1/221, this means you are more likely to get 2 aces in a row if you put your first ace back in the deck before you draw your second card. But you knew that!

Probabilities as percents:

Sometimes probabilities are given as percentages. This is just changing the fraction we have been talking about into hundredths. So 1/2 becomes 50/100 or 50%. To change a fraction to a percent:
  1. Use your calculator to divide the numerator by the denominator
  2. Multiply the result by 100

So, to convert a probability of 2/7 to a percent:
  1. Divide the numerator by the denominator: 2 divided by 7 = .2857
  2. Multiply by 100: .2857 x 100 = 28.57 = 28.57%