Question

In: Statistics and Probability

The following events are considered for a throw with two dice: A: The total of eyes...

The following events are considered for a throw with two dice:
A: The total of eyes is greater than 7.
B: Exactly one of the two numbers is a 5.
C: No 1 is rolled.
a) Calculate the probabilities
P (A), P (B), P (C), P (A ∩ B), P (A ∩ C), P (B ∩ C), P (A ∪ B), P (A | B), P (A | C), P (C | A), P (B | C).
b) Are events A and B independent or disjoint?

Solutions

Expert Solution

(a) Total number of cases is 36. favorable cases are shown in bold.

(1,1)    2 (2,1)    3 (3,1)    4 (4,1)    5 (5,1)    6 (6,1)    7
(1,2)    3 (2,2)    4 (3,2)    5 (4,2)    6 (5,2)    7 (6,2)    8
(1,3)    4 (2,3)    5 (3,3)    6 (4,3)    7 (5,3)    8 (6,3)    9
(1,4)    5 (2,4)    6 (3,4)    7 (4,4)    8 (5,4)    9 (6,4) 10
(1,5)    6 (2,5)    7 (3,5)    8 (4,5)    9 (5,5) 10 (6,5) 11
(1,6)    7 (2,6)    8 (3,6)    9 (4,6) 10 (5,6) 11 (6,6) 12

P(A) = 15/36

(1,1)    2 (2,1)    3 (3,1)    4 (4,1)    5 (5,1)    6 (6,1)    7
(1,2)    3 (2,2)    4 (3,2)    5 (4,2)    6 (5,2)    7 (6,2)    8
(1,3)    4 (2,3)    5 (3,3)    6 (4,3)    7 (5,3)    8 (6,3)    9
(1,4)    5 (2,4)    6 (3,4)    7 (4,4)    8 (5,4)    9 (6,4) 10
(1,5)    6 (2,5)    7 (3,5)    8 (4,5)    9 (5,5) 10 (6,5) 11
(1,6)    7 (2,6)    8 (3,6)    9 (4,6) 10 (5,6) 11 (6,6) 12

P(B) = 10/36 = 5/18

(1,1)    2 (2,1)    3 (3,1)    4 (4,1)    5 (5,1)    6 (6,1)    7
(1,2)    3 (2,2)    4 (3,2)    5 (4,2)    6 (5,2)    7 (6,2)    8
(1,3)    4 (2,3)    5 (3,3)    6 (4,3)    7 (5,3)    8 (6,3)    9
(1,4)    5 (2,4)    6 (3,4)    7 (4,4)    8 (5,4)    9 (6,4) 10
(1,5)    6 (2,5)    7 (3,5)    8 (4,5)    9 (5,5) 10 (6,5) 11
(1,6)    7 (2,6)    8 (3,6)    9 (4,6) 10 (5,6) 11 (6,6) 12

P(C) = 11/36

(1,1)    2 (2,1)    3 (3,1)    4 (4,1)    5 (5,1)    6 (6,1)    7
(1,2)    3 (2,2)    4 (3,2)    5 (4,2)    6 (5,2)    7 (6,2)    8
(1,3)    4 (2,3)    5 (3,3)    6 (4,3)    7 (5,3)    8 (6,3)    9
(1,4)    5 (2,4)    6 (3,4)    7 (4,4)    8 (5,4)    9 (6,4) 10
(1,5)    6 (2,5)    7 (3,5)    8 (4,5)    9 (5,5) 10 (6,5) 11
(1,6)    7 (2,6)    8 (3,6)    9 (4,6) 10 (5,6) 11 (6,6) 12

P(A ∩ B) = 6/36 = 1/6

(1,1)    2 (2,1)    3 (3,1)    4 (4,1)    5 (5,1)    6 (6,1)    7
(1,2)    3 (2,2)    4 (3,2)    5 (4,2)    6 (5,2)    7 (6,2)    8
(1,3)    4 (2,3)    5 (3,3)    6 (4,3)    7 (5,3)    8 (6,3)    9
(1,4)    5 (2,4)    6 (3,4)    7 (4,4)    8 (5,4)    9 (6,4) 10
(1,5)    6 (2,5)    7 (3,5)    8 (4,5)    9 (5,5) 10 (6,5) 11
(1,6)    7 (2,6)    8 (3,6)    9 (4,6) 10 (5,6) 11 (6,6) 12

P (A ∩ C) = 0/36 = 0

(1,1)    2 (2,1)    3 (3,1)    4 (4,1)    5 (5,1)    6 (6,1)    7
(1,2)    3 (2,2)    4 (3,2)    5 (4,2)    6 (5,2)    7 (6,2)    8
(1,3)    4 (2,3)    5 (3,3)    6 (4,3)    7 (5,3)    8 (6,3)    9
(1,4)    5 (2,4)    6 (3,4)    7 (4,4)    8 (5,4)    9 (6,4) 10
(1,5)    6 (2,5)    7 (3,5)    8 (4,5)    9 (5,5) 10 (6,5) 11
(1,6)    7 (2,6)    8 (3,6)    9 (4,6) 10 (5,6) 11 (6,6) 12

P (B ∩ C) = 2/36 = 1/18

P (A ∪ B) = P(A) + P(B) - P(A ∩ B) = 24/36 = 2/3

P (A | B) = P(A ∩ B) / P(B) = 6/10 = 0.6

P (A | C) = P(A ∩ C) / P(C) = 0

P (C | A) = P(C ∩ A) / P(A) = P(A ∩ C) / P(A) = 0

P (B | C) = P (B ∩ C) / P(C) = 2/11

(b)

P(A ∩ B) = 6/36 = 1/6 is not equal to 0, hence A and B are not disjoint.

Also P(A)*P(B) = (15*10)/(36*36) is not equal to P(A ∩ B) = 1/6, hence A and B are not independent.

So, the events A and B are neither independent nor disjoint.

orchestra answered 2 years ago
Next > < Previous

Related Solutions

For a roll with two dice, the following events are considered: A: The sum of eyes...
For a roll with two dice, the following events are considered: A: The sum of eyes is greater than 7. B: Exactly one of the two numbers is a 5. C: No 1 is rolled. a) Calculate the probabilities P (A), P (B), P (C), P (A ∩ B), P (A ∩ C), P (B ∩ C), P (A ∪ B), P (A | B), P (A | C), P (C | A), P (B | C). b) Are events...
For a roll with two dice, the following events are considered: A: The sum of eyes...
For a roll with two dice, the following events are considered: A: The sum of eyes is greater than 7. B: Exactly one of the two numbers is a 5. C: No 1 is rolled. a) Calculate the probabilities P (A), P (B), P (C), P (A ∩ B), P (A ∩ C), P (B ∩ C), P (A ∪ B), P (A | B), P (A | C), P (C | A), P (B | C). b) Are events...
Throw two dice. If the sum of the two dice is 6 or more, you win...
Throw two dice. If the sum of the two dice is 6 or more, you win $43. If not, you pay me $117 . Step 2 of 2 : If you played this game 907 times how much would you expect to win or lose? Round your answer to two decimal places. Losses must be entered as negative.
Throw two dice. If the sum of the two dice is 6or more, you win $9.If...
Throw two dice. If the sum of the two dice is 6or more, you win $9.If not, you pay me $30. Step 1 of 2: Find the expected value of the proposition. Round your answer to two decimal places. Losses must be expressed as negative values.
Consider the experiment of rolling two dice and the following events:     A: 'The sum of the...
Consider the experiment of rolling two dice and the following events:     A: 'The sum of the dice is 8' and  B: 'The first die is an odd number' and C:  "The difference (absolute value) of the dice is 2" Find  (a)  p(A and B) (HINT: You cannot assume these are independent events.)        (b)  p(A or B)         (c)  Are A and B mutually exclusive events? Explain. (d)   Are A and B independent events? Explain. (e)   Are B and C independent events? Explain.
In order to win a game, a player must throw two fair dice and the sum...
In order to win a game, a player must throw two fair dice and the sum of the dice needs to be either 4 or less or 10 or more for the player to win. What is the probability that the sum of the dice is 4 or less? What is the probability that the sum of the dice is 10 or more? What is the probability that the player will win the game?
Let’s have a dice and a coin. Randomly throw the dice once and the coin twice....
Let’s have a dice and a coin. Randomly throw the dice once and the coin twice. Determine the space of elementary events and P (A∪B) and check the independence of events A and B, if event A consists in the fact that on dice we have odd number, B that on coin we have heads twice and on dice we have number lower than 3.
Let’s have a dice and a coin. Randomly throw the dice once and the coin twice....
Let’s have a dice and a coin. Randomly throw the dice once and the coin twice. Determine the space of elementary events and P (A∪B) and check the independence of events A and B, if event A consists in the fact that on dice we have even number, B that on coin we have heads twice and on dice we have number lower than 3.
Game of dice - fair dice: If I throw a number >= 5 I win. If...
Game of dice - fair dice: If I throw a number >= 5 I win. If he throws a number =< 4 he wins. I throw the first dice. Given that he loses(throws a number >4) what is the probability of me winning ? What is the expected number of throws before either of us wins?
Take 3 dice and throw these dice 30 times. Let X be the sum of the...
Take 3 dice and throw these dice 30 times. Let X be the sum of the number of dots on upper faces of the dice. Obtain probability distribution of X. Also find mean and variance.
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT