In-class review for Test 1 SECTONS 7.4, 7.5 (Spring 2013)
1. Solution:
2. Draw a marble at random from a box containing
2 Yellow, 5 White & 6 Blue marbles.
a) What is the probability of drawing a Yellow marble?
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b) What are the odds of drawing a Yellow marble?
ODDS: m : n -> wanted : not wanted [and m + n = total]
Here m=2 (want a Yellow) and n=11 (don’t want any of the others)
Answer -> 2 : 11 (can also be written 2 to 11 or 2/11)
c) What are the odds in favor of drawing a Blue marble?
ODDS: m : n -> wanted : not wanted [and m + n = total]
Here m=6 (want a Blue) and n=7 (don’t want any of the others)
Answer -> 6 : 7 (can also be written 6 to 7 or 6/7)
d) What are the odds against drawing a Blue marble?
ODDS AGAINST something are just the reciprocal of ODDS IN FAVOR of it
So, from c) above, the odds in favor of Blue are 6 : 7.
Thus, the odds against drawing a Blue marble are 7 : 6.
A pair of dice is rolled.
What are the odds against rolling a sum of 7?
Sample Space:
11 12 13 14 15 16
21 22 23 24 25 26
31 32 33 34 35 36
41 42 43 44 45 46
51 52 53 54 55 56
61 62 63 64 65 66
ODDS IN FAVOR: m : n -> wanted : not wanted [and m + n = total]
Here m=6 (want sum of 7) and n=30 (don’t want any of the other sums)
So, odds in favor of a sum of 7 is 6 : 30 which reduces to 1 : 5
And the odds against a sum of 7 is 30 : 6 which reduces to 5 : 1
4. If the odds in favor of Chris winning the election are 5 : 4,
then what is the probability that Chris wins?
ODDS IN FAVOR: m : n -> wanted : not wanted [and m + n = total]
Here m=5 (wanted: Chris wins) and n=4 (not wanted: Chris loses)
So Total = m + n = 5 + 4 = 9
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5. Two cards are drawn without replacement from an
ordinary deck of cards. What is the probability that the
second card is not a Spade given that the first card is not
a Spade?
Before 1st draw: 52 cards -> 13 Spades & 39 non-Spades
After 1st draw: 51 cards -> 13 Spades & 38 non-Spades
(reduced sample space)
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6. The table shows the results of a restaurant survey. Find the probability that
service was good given that the meal was lunch.
MEAL
Lunch
Dinner
TOTAL
GOOD
SERVICE
34
42
76
POOR
SERVICE
21
39
60
TOTAL
55
81
136
REDUCED SAMPLE SPACE
(“given that the meal was lunch”)
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7. Suppose one of the following is chosen at random:
1 2 3 4 5 6
What is the probability the number chosen is even
given that it is greater than 5 ?
Original Sample Space: { 1 , 2 , 3 , 4 , 5 , 6 }
The condition “greater than 5” reduces the sample space to: { 6 }
Notice that 6 is an even number and is the only element in the reduced
sample space so,
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8. Suppose a single fair die is rolled.
Find the probability that it is an even number
given that it is a 1.
Original Sample Space: { 1 , 2 , 3 , 4 , 5 , 6 }
The condition “it is a 1” reduces the sample space to: { 1 }
Notice that 1 is an odd number and is the only element in the reduced
sample space so,
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9. Two dice are rolled.
What is the probability that the sum on the dice is 3 if the first die is a 2 ?
Start off with the unrestricted sample space and apply the given “the first die
is a 2” to get the reduced sample space:
11 12 13 14 15 16
21 22 23 24 25 26
31 32 33 34 35 36
41 42 43 44 45 46
51 52 53 54 55 56
61 62 63 64 65 66
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10. Let A and B be independent events with
( )
and ( )
Find (
.
) and (
For independent events: (
So, (
).
)
( )
( )
)
( )
)
The Union Rule for Probability is (
(
)
( )
(
).
11. Two cards are drawn without replacement from an ordinary deck. Find
the probability that the second card is not red given that the first card is
red.
Before 1st draw: 52 cards -> 26 Red & 26 non-Red
After 1st draw: 51 cards -> 25 Red & 26 non-Red
(reduced sample space: given 1st card is Red)
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