Section 4.5 “AND problems”
The AND problems discussed in this section
involve repeating an experiment twice.
An “AND” probability problem requires
obtaining a favorable outcome in each of the
given events. This is the formula we will use to
solve “AND” probabilities
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∗ 𝑃(𝐵)
Example: Two cards are selected with
replacement from a standard deck of cards.
Determine the probability that two queens will
be selected. We consider this an “AND”
problem as we need to get a queen on the first
selection AND on the second selection.
I need to be able to compute two probabilities
to answer this question.
P(queen 1st) = 4/52 as there are 4 queens in a
deck of 52 cards
P(queen 2nd) = 4/52 since this is a with
replacement problem there are still 4 queens of
the 52 cards that can be selected.
P  2 queens   P  queen1  P  queen2 
Answer: 1/169
4
52
×
4
52
=
1
169
= .59%
Example: Two cards are selected without
replacement from a standard deck of cards.
Determine the probability that two queens will
be selected.
I need to be able to compute two probabilities
to answer this question.
P(queen 1st) = 4/52 as there are 4 queens in a
deck of 52 cards
P(queen 2nd) = 3/51 This is a without
replacement problem only 3 queens left as one
of them has been selected already. There are
only 51 cards left as the queen that was
selected first is no longer in the deck
P  2 queens   P  queen1  P  queen2 
4
52
3
1
× =
= .45%
51
221
Answer: 1/221
Definition: Event A and Event B are called
independent events if the occurrence of either
event in no way affects the probability of
occurrence of the other event.
Rolling a die and tossing a coin are examples of
independent events. This is because if a coin
comes up heads or tails it won’t affect the
number that comes up on a dice.
It’s a little trickier to determine if events are
independent with the card scenario we just
covered.
In our 2 Queens example (the first one WITH
replacement) is an example of an
INDEPENDENT events problem as the
probability of getting a queen on the second
selection did not change do to the first
selection.
BUT the second 2 Queens example (the one
WITHOUT replacement) is an example of a
DEPENDENT events problem because the
probability of selecting the second queen was
affected by removing the first queen selected
form the deck.
Homework # 1 – 10: Two cards are selected at
random from a deck of 52 cards WITHOUT
replacement. Find the requested probabilities,
(write your answer as a reduced fraction.)
1) Both cards are red
2) Both cards are black
3) The first card is a 7 and the second card is a
4
4) The first card is a queen and the second card
is a 3
5) Both cards are fives
6) Both cards are face cards
7) The first card is red and the second card is
black
8) The first card is a heart and the second card
is a spade
9) The first card is not a seven and the second
card is a seven
10) The first card is not a four and the second
card is a four
Homework # 11 – 20: Two cards are selected at
random from a deck of 52 cards WITH
replacement. Find the requested probabilities
(write your answer as a reduced fraction)
11) Both cards are hearts
12) Both cards are spades
13) The first card is a six and the second card is
black
14) The first card is black and the second card
is a six
15) The first card is a face card and the second
card is not a face card
16) Both cards are face cards
17) Neither card is black
18) Neither card is a spade
19) The first card is a queen and the second
card is a seven
20) The first card is not a queen and the
second card is not a seven
Homework # 21-24: The spinner below is spun
twice, find the requested probabilities (write
your answer as a reduced fraction.)
21) Both spins are yellow
22) Both spins are purple
23) The first spin is green and the second is not
red
24) The first spin is not yellow and the second
spin is red
Homework # 25-28: The spinner below is spun
three times, find the requested probabilities
(write your answer as a reduced fraction.)
25) The spins in order are green, yellow the
blue
26) The spins in order are green, green and
yellow
27) Green is spun each time
28) Yellow is spun each time
Homework #29 – 32: A coin is tossed then a
dice is rolled, find the requested probabilities
(write your answer as a reduced fraction.)
29) The coin is a head and the dice is a 4
30) The coin is a tail and the dice is not a 4
31) The coin is not a head and the dice is an
even number
32) The coin is not a tail and the dice is greater
than 2
Homework #33-34: A sample of 30 women
who recently had a home built yielded the
following information about their builder:
Number of women
19
6
5
Would you
recommend builder to
a friend
Yes
No
Not sure
Three women who provided information for
the table were selected at random. Find the
probability that:
33) The first would not recommend her home
builder, but the second and third would
recommend their home builder.
34) The first would recommend her home
builder, but the second and third would not
recommend their home builder.
Homework #35-38
Two marbles are drawn from a bag that
contains 5 blue, 3 red and 2 purple marbles
without replacement. Find the requested
probabilities
35) Both are blue
36) Both are red
37) The first is purple and the second is not
blue
38) The first is purple and the second is red
Homework #39-42
Two marbles are drawn from a bag that
contains 5 blue, 3 red and 2 purple marbles
with replacement. Find the requested
probabilities
39) Both are blue
40) Both are red
41) The first is purple and the second is not
blue
42) The first is not purple and the second is red
Answers, with a bit of work:
1)
26
52
11)
21)
29)
39)
25
25
∗ 51 = 102 3)
13
13
4
1
5
1
2
1
1
∗ 5 = 25
1
1
∗ 6 = 12
5
10
5
31)
1
∗ 10 = 4
5
1
2
41)
4
26
1
4
4
∗ 5 = 25 25)
3
1
∗6=4
2
10
5
33)
1
∗ 10 = 10
3
1
2
1
∗ 51 = 221 7)
52
∗ = 26 15)
52 52
1
23)
4
∗ 51 = 663 5)
4
∗ = 16 13)
52 52
1
4
52
4
40
1
6
52
10
∗ = 169 17)
52 52
1
1
13
26
26
∗ 29 ∗ 28 = 2030 35)
10
19
18
171
2
1
48
52
∗ = 4 19)
52 52
1
27)
1
26
∗ 51 = 51 9)
1
∗ 4 ∗ 4 = 32
30
26
4
16
∗ 51 = 221
4
4
1
∗ = 169
52 52
1
∗2∗2=8
5
4
2
∗ 9 = 9 37)
2
10
4
4
∗ 9 = 45