16.4 Probability Problems Solved with Combinations
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Transcript 16.4 Probability Problems Solved with Combinations
16.4 Probability Problems Solved
with
Combinations
16.4 Warm Up
Three cards are drawn from a well-shuffled standard deck of 52 cards,
one after the other and without replacement.
1. Find the probability of drawing
a) all clubs 13 12 11 11
b) no clubs 39 38 37 703
52 51 50
52 51 50
850
1700
c) exactly one club (hint: the club can occur on the 1st, 2nd or 3rd
drawing) 13 39 38 741
3
52
51
50 1700
2. Evaluate:
a)
c)
13 C3
52 C3
11
850
13 C139 C2
52 C3
b)
39 C3
52 C3
703
1700
741
1700
3. Compare your answers to exercises 1 and 2. What do you notice?
They are the same!
4. Write and evaluate an expression using combinations to find the
probability of getting exactly 2 clubs. 13 C2 39 C1 117
52 C3
850
16.4: Solving Probability Problems w/ Combinations
Recall the definition of probability:
If E is an event from a sample space S of equally likely outcomes,
the probability of event E is:
P (E )
n(E ) # favorable
n(S )
# total
0 P(E) 1.
• Counting techniques can be used to determine the number of
favorable and total number.
• Combinations automatically handle situations where multiple cases
can occur (e.g., 3 cards with one ace: ANN, NAN & NNA).
Example: Three marbles are picked at random from a bag containing 4
red marbles and 5 white marbles. What is the probability that
1
4 C3 5 C0
a) all 3 are red
b) 2 red marbles 4 C2 5 C1 5
9 C3
c) 1 marble is red
9 C3
21
3+0=3
10
4 C15 C2
21
9 C3
d) no red marbles
4 C0 5 C3
9 C3
1+2=3
Notice that the number selected in the numerator is always 3.
14
2+1=3
5
42
0+3=3
Examples
1) Five cards are randomly chosen from a standard deck of 52 cards.
Find the probability that the following are chosen
a) all 4 aces 48 C14 C4
b) No aces
35,673
1
48 C5
52 C5
52 C5
54,145
c) exactly 4 diamonds
13 C4 39 C1
52 C5
54,145
d) four aces and one jack
4 C4 4 C1
143
13,328
52 C5
1
649,740
e) at least one ace
1 – P(no aces) 1
48 C5
52 C5
18,472
54,145
2) Three cards are dealt. Find the probability of getting either one ace or
two aces.
48
4 C148 C2
4 C2 48 C1
52 C3
52 C3
221
3) A carton contains 200 batteries, of which 5 are defective. If a random
sample of 5 batteries is chosen, what is the probability that at least
one is defective?
P(at least one defective) = 1 – P(no defective) 1
195 C5
200 C5
1 0.12 0.88
OR P(at least one defective) = P(1 def) + P(2 def) + P(3 def) + P(4 def) + P(5 def)
much more complicated…
4) Thirteen cards are dealt from a well shuffled standard card deck.
What is the probability of getting:
a) all cards from the same suit
b) 7 spades, 3 hearts, and 3 clubs
P(all any 1 suit) = P(all ♠) + P(all ♣) +
P(all ♥) + P(all ♦)
13 C13
4
= 4·P(all ♠)
52 C13
13 C7 13 C3 13 C3
52 C13
2.21X10 4
6.30 X 10 12
c) all of the 12 face cards
12 C12 40 C1
52 C13
6.20 X 1011
e) at least one diamond
P(at least one diamond) = 1 – P(no diamonds) 1
39 C5
52 C5
1 0.22 0.78
Should counting always be used?
6. A die is rolled twice
a) Find the probability that 2 sixes are rolled
2
1
1
6
36
b) Find the probability that the face value is greater than 4 and that
the second is 2.
2 1 1
6 6 18
These are easier to do using probabilities and multiplication.