quiz 3
in order to receive credit you will have to provide adequate explanation as to how you arrived at an answer. answers given with no supporting work will not receive credit. answers that are correct but are accompanied by incorrect steps that would imply a different answer than that given will also not be awarded credit.
1. (12) suppose that you roll two dice, one after the other, and record the number of dots on the face of each die
(a) (4) list all of the 36 possible outcomes of this experiment.
(b) (2) what is the probability that the number of dots on the faces of the dice are all the same?
(c) (2) what is the probability that the number of dots on the faces of the dice are not all the same?
(d) (2) what is the probability that the sum of the dots on the faces of the dice is less than or equal to 5?
(e) (2) what is the probability that the sum of the dots on the faces of the dices is greater than or equal to 8.
2. (12) suppose that e and f are events in a sample - space s with (p(ecap f) = 0.23), (p(f) = 0.65), (p(e) = 0.55).
(a) (6) draw a venn diagram with the events e and f as circles and label the probabilities of the four events (ecap f), (ecap f), (ecap f), ((ecup f)).
(b) (2) (p(e))
(c) (2) (p(ecup f))
(d) (2) (p(ecup f))
3. (8) suppose that tomorrows weather forecast calls for rain with probability 0.35, a daily high temperature above 40 with probability 0.70, and rain with a daily high temperature of 40 degrees or below with probability 0.12.
(a) (4) let t be the event that daily high temperature is greater than 40, and let r be the event that it will rain. draw a venn diagram with these events as circles and label the probabilities of the events (tcap r), (tcap r), (tcap r), ((tcup r)).
(b) (2) what is the probability that it will not rain and the daily high temperature will be above 40?
(c) (2) what is the probability that is will not rain and the daily high temperature will be 40 degrees or below.?

Question

Accepted Answer

### 1.
#### (a)
# Explanation:
## Step1: List all pairs
(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),
(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)

#### (b)
# Explanation:
## Step1: Identify favorable outcomes
Favorable outcomes are (1,1),(2,2),(3,3),(4,4),(5,5),(6,6), so there are 6.
## Step2: Calculate probability
The probability $P=\frac{6}{36}=\frac{1}{6}$
# Answer:
$\frac{1}{6}$

#### (c)
# Explanation:
## Step1: Use complementary - probability
The probability that the numbers are not all the same is the complement of the event that the numbers are the same.
## Step2: Calculate probability
$P = 1-\frac{1}{6}=\frac{5}{6}$
# Answer:
$\frac{5}{6}$

#### (d)
# Explanation:
## Step1: List favorable outcomes
Favorable outcomes for sum $\leq5$ are (1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(3,1),(3,2),(4,1), so there are 10.
## Step2: Calculate probability
$P=\frac{10}{36}=\frac{5}{18}$
# Answer:
$\frac{5}{18}$

#### (e)
# Explanation:
## Step1: List favorable outcomes
Favorable outcomes for sum $\geq8$ are (2,6),(3,5),(3,6),(4,4),(4,5),(4,6),(5,3),(5,4),(5,5),(5,6),(6,2),(6,3),(6,4),(6,5),(6,6), so there are 15.
## Step2: Calculate probability
$P=\frac{15}{36}=\frac{5}{12}$
# Answer:
$\frac{5}{12}$

### 2.
#### (a)
1. First, find $P(F)=1 - P(F')=1 - 0.65 = 0.35$ and $P(E)=1 - P(E')=1 - 0.55 = 0.45$.
2. Then, use the formula $P(E)=P(E\cap F)+P(E\cap F')$. So, $P(E\cap F)=P(E)-P(E\cap F')=0.45 - 0.23 = 0.22$.
3. $P(E'\cap F)=P(F)-P(E\cap F)=0.35 - 0.22 = 0.13$.
4. $P((E\cup F)')=1 - P(E\cup F)$. And $P(E\cup F)=P(E)+P(F)-P(E\cap F)=0.45 + 0.35-0.22 = 0.58$, so $P((E\cup F)')=1 - 0.58 = 0.42$.
Draw two - overlapping circles labeled E and F. Label the region $E\cap F'$ with 0.23, $E\cap F$ with 0.22, $E'\cap F$ with 0.13 and $(E\cup F)'$ with 0.42.

#### (b)
# Explanation:
## Step1: Use the complement rule
$P(E)=1 - P(E')$
## Step2: Calculate
$P(E)=1 - 0.55 = 0.45$
# Answer:
$0.45$

#### (c)
# Explanation:
## Step1: Use the addition rule
$P(E\cup F)=P(E)+P(F)-P(E\cap F)$
## Step2: Calculate
We know $P(E) = 0.45$, $P(F)=0.35$ and $P(E\cap F)=0.22$. So $P(E\cup F)=0.45 + 0.35-0.22 = 0.58$
# Answer:
$0.58$

#### (d)
# Explanation:
## Step1: Use De - Morgan's law
$P(E'\cup F')=P((E\cap F)')=1 - P(E\cap F)$
## Step2: Calculate
$P(E\cap F)=0.22$, so $P(E'\cup F')=1 - 0.22 = 0.78$
# Answer:
$0.78$

### 3.
#### (a)
1. We know $P(R) = 0.35$, $P(T)=0.70$ and $P(R\cap T')=0.12$.
2. $P(R\cap T)=P(R)-P(R\cap T')=0.35 - 0.12 = 0.23$.
3. $P(T\cap R')=P(T)-P(T\cap R)=0.70 - 0.23 = 0.47$.
4. $P((T\cup R)')=1 - P(T\cup R)$. And $P(T\cup R)=P(T)+P(R)-P(T\cap R)=0.70 + 0.35-0.23 = 0.82$, so $P((T\cup R)')=1 - 0.82 = 0.18$.
Draw two - overlapping circles labeled T and R. Label the region $T\cap R'$ with 0.47, $T\cap R$ with 0.23, $T'\cap R$ with 0.12 and $(T\cup R)'$ with 0.18.

#### (b)
# Explanation:
## Step1: Identify the event
The event is $T\cap R'$.
## Step2: Get probability from part (a)
$P(T\cap R') = 0.47$
# Answer:
$0.47$

#### (c)
# Explanation:
## Step1: Identify the event
The event is $T'\cap R'$.
## Step2: Get probability from part (a)
$P((T\cup R)')=P(T'\cap R') = 0.18$
# Answer:
$0.18$

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QUESTION IMAGE

Question

Explanation:

1.

(a)

Step1: List all pairs

(b)

Step1: Identify favorable outcomes

Step2: Calculate probability

Step1: Use complementary - probability

Step2: Calculate probability

Answer:

(c)