Download Probability With A Deck Of Cards

Probability and Odds
A. Many times we are given (or can find) the probability but the question asks for the odds. So we must change probability to odds.
B. First, let's look at the difference between probability and odds:
probability = # of desirable outcomes /# of possible outcomes (always a fraction in simplest form)
odds in favor = # of desirable outcomes: # of undesirable outcomes (always written using a colon, not a fraction)
Odds against = # of undesirable outcomes :# of desirable outcomes (always written using a colon, not a fraction)
C. One way of looking at this is to look at a simple example involving a coin.
1. The probability of flipping a coin and getting heads is 1/2 because there are two possibilities [heads or tails].
2. The odds of getting heads is 1/1 which means the chances of getting heads and the chances of getting tails are equal.
D. Let's look at how to change from one to the other.
Probability -> Odds:
1. You should note that the numerator (first #) is the same in both probability and odds, so the numerator of the answer will be
the same.
2. To find the denominator (or the second #) we should subtract the numerator from the denominator.
Ex [1] If the probability of winning is 7/15, then the odds in favor of winning is __________?
a. The first # is the same which is 7.
b. The second # is: 15-7 = 8.
c. The answer is 7:8
Ex [2] If the probability of losing is 6/17, then the odds against losing is __________?
a.
b.
c.
d.
We must first find the odds in favor of losing, then reverse the answer
The first # is the same which is 6.
The second # is: 17-6 = 11.
The odds in favor of losing is 6:11 which means the odds against losing is 11:6
Odds -> Probability
1. Again, the numerator (first #) is the same in both probability and odds, so the numerator of the answer will be the same.
2. To find the denominator (the number of possible outcomes) you simple add the numerator and denominator.
Ex [1] If the odds of winning is 6:7 then the probability of winning is __________?
a. The numerator is the same which is 6.
b. The denominator is: 6+7 = 13.
c. The answer is 6/13
Ex [2] If the odds of losing is 9/10 then the probability of winning is __________?
a. Notice we are changing from losing to winning, not from losing to losing. So we have to first find the
probability of losing then subtract that value from 1.
b. The numerator is the same which is 9.
c. The denominator is: 9+10 = 19.
d. The probability of losing is 9/19 so the probability of winning is 10/19. (Subtract 1-9/19).
Probability With A Deck Of Cards:
A. Sometimes a number sense test will ask for the probability of drawing a particular card from a deck of 52 cards, or they might say from a
standard deck (which is the same thing). There are many patterns that develop.
B. If the question is talking about more than one card, they might use the phrase "with replacement" or "without replacement". If no phrase is
used, it is assumed "without replacement". If additional cards are drawn "with replacement", then the card is put back before drawing the
next card, otherwise it is not.
C. If drawing only one card:
1. The probability of drawing a specific card is 1/52.
2. The probability of drawing a specific number (or specific face card) is 4/52 or 1/13.
3. The probability of drawing a specific suit (club, spade, diamond, or heart) is 13/52 or 1/4.
D. If drawing more than one card with replacement:
1. The probability of drawing a specific card each time is 1/52n where n is the number of draws.
2. The probability of drawing a specific number (or specific face card) each time is 1/13n.
3. The probability of drawing a specific suit each time is 1/4n.
E. If drawing more than one card without replacement (hardest):
1. The probability of drawing a specific number (or specific face card) each time is 4/52 x 3/51 x 2/50 or 1/13 x 1/17 x 1/25. Note: This
is if you wanted to draw 3 specific numbers. If you wanted 2, it would only be the first 2 terms multiplied by each other.
2. The probability of drawing a specific suit each time is 13/52 x 12/51 x 11/50 x ... or 1/4 x 4/17 x 11/50 x ...
F. Examples
Ex [1] A card is drawn at random from a standard deck of cards. The probability it is a spade is __________.
a. The answer is 13/52 or 1/4.
Ex [2] 2 cards are drawn from a deck of 52 cards, with replacement. What is the probability they were both the King of hearts?
a. The first draw, the probability is 1/52.
b. The second draw, the probability is still 1/52.
c. The answer is 1/522 or 1/2704.
Ex [3] Find the probability of drawing 3 Aces from a standard deck of cards.
a. Since the question does not specify "with replacement" we can assume it is "without replacement".
b. The first draw the probability is 4/52 or 1/13.
c. The second draw the probability is 3/51 or 1/17.
d. The third draw the probability is 2/50 or 1/25.
e. The answer is the product of all of these.
f. Since the numerator is 1, we only have to focus on the denominator or 13 x 17 x 25.
i. The answer is 1/5525.
Probability With Dice:
A. Since 1 die only has 6 sides, the chances of rolling a specific number is only 1/6, because there are only 6 possibilities. When working
with 2 dice, there are 36 possible outcomes which is calculated by 62.
B. Below is a chart that will help explain all 36 possible outcomes. On the left will be the sum of the two dice and on the right, in the form of
a:b, is a possible combination of 2 dice. The last number in () is the number of different combinations that are possible to reach the sum.
2 -> 1:1 (1)
3 -> 2:1, 1:2 (2)
4 -> 1:3, 3:1, 2:2 (3)
5 -> 1:4, 4:1, 2:3, 3:2 (4)
6 -> 1:5, 5:1, 2:4, 4:2, 3:3 (5)
7 -> 1:6, 6:1, 2:5, 5:2, 3:4, 4:3 (6)
8 -> 2:6, 6:2, 3:5, 5:3, 4:4 (5)
9 -> 3:6, 6:3, 4:5, 5:4 (4)
10 -> 4:6, 6:4, 5:5 (3)
11 -> 5:6, 6:5 (2)
12 -> 6:6 (1)
D. What does this chart show? This chart shows the probability for rolling a particular sum given 2 dice. In other words, the probability for
rolling a sum of 8 is 5/36. The probability for rolling a sum of 12 is 1/36.
E. Examples:
Ex [1] A single die is rolled. What is the probability that the number is divisible by 3?
a. For a problem of this type, we need to know what numbers are divisible by 3. The only numbers that are, are 3 and 6.
So we need to add each probability.
b. The probability you rolled a 3 is 1/6.
c. The probability you rolled a 6 is 1/6.
d. 1/6 + 1/6 = 2/6 = 1/3.
e. The answer is 1/3.
Ex [2] A pair of dice are rolled, find the probability that the sum is greater than 9.
a.
b.
c.
d.
e.
f.
For a problem of this type, we need to know the probability for the sums of 10, 11, and 12. (Notice: 9 is not included)
The probability the sum is 10 is 3/36. (Don't reduce)
The probability the sum is 11 is 2/36.
The probability the sum is 12 is 1/36.
Adding these together we get: 3+2+1/36 = 6/36 = 1/6.
The answer is 1/6.
Ex [3] A pair of dice are rolled. Find the probability that the sum is a multiple of 4.
a. For a problem of this type, we need to know what possible sums there are that are divisible by 4. They are 4, 8, and
12. So we need to add each probability.
b. The probability you rolled a sum of 4 is 3/36. (Don't reduce)
c. The probability you rolled a sum of 8 is 5/36.
d. The probability you rolled a sum of 12 is 1/36.
e. Adding all of these together we get: 3+5+1/36 = 9/36 = 1/4.