### Probability Exercise 02

## Question:

Two dice are thrown. The events A,B and C are as follows: A : getting an even number on the first die B : getting an odd number on the first die C : getting the sum of the numbers on the dice ≤5 State true or false : (give reason for your answer) (i) A and B are mutually exclusive (ii) A and B are mutually exclusive and exhaustive (iii) A=B′ (iv) A and C are mutually exclusive (v) A and B′ are mutually exclusive (vi) A′,B′,C are mutually exclusive and exhaustive

## Answer:

(i) True. A and B are mutually exclusive because they cannot occur at the same time.

(ii) False. A and B are mutually exclusive, but not exhaustive because C is also a possible event.

(iii) False. A does not equal B’, because B’ is the complement of B, which is getting an odd number on the first die.

(iv) False. A and C are not mutually exclusive because it is possible to get an even number on the first die and a sum of the numbers on the dice ≤5.

(v) True. A and B’ are mutually exclusive because they cannot occur at the same time.

(vi) False. A’,B’,C are not mutually exclusive and exhaustive because it is possible to get an even number on the first die and a sum of the numbers on the dice ≤5.

## Question:

Three coins are tossed describe (i) Two events which are mutually exclusive (ii) Three events which are mutually exclusive and exhaustive (iii) Two events which are not mutually exclusive (iv) Two events which are mutually exclusive but not exhaustive (v) Three events which are mutually exclusive but not exhaustive

## Answer:

(i) Two events which are mutually exclusive: Getting heads on the first coin and getting tails on the first coin.

(ii) Three events which are mutually exclusive and exhaustive: Getting heads on the first coin, getting tails on the first coin, and getting a combination of heads and tails on the first coin.

(iii) Two events which are not mutually exclusive: Getting heads on the first two coins and getting heads on all three coins.

(iv) Two events which are mutually exclusive but not exhaustive: Getting heads on the first coin and getting tails on the second coin.

(v) Three events which are mutually exclusive but not exhaustive: Getting heads on the first coin, getting tails on the second coin, and getting a combination of heads and tails on the third coin.

## Question:

Two dice are thrown. The events A, B and C are as follows: A: getting an even number on the first die. B: getting an odd number on the first die. C: getting the sum of the numbers on the dice ≤ 5 Describe the events (i) A’ (ii) not B (iii) A or B (iv) A and B (v) A but not C (vi) B or C (vii) B and C (viii) A∩ B’∩ C'

## Answer:

(i) Event A: Getting an even number on the first die.

(ii) Event not B: Not getting an odd number on the first die.

(iii) Event A or B: Getting an even number on the first die or getting an odd number on the first die.

(iv) Event A and B: Getting an even number on the first die and getting an odd number on the first die.

(v) Event A but not C: Getting an even number on the first die but not getting the sum of the numbers on the dice ≤ 5.

(vi) Event B or C: Getting an odd number on the first die or getting the sum of the numbers on the dice ≤ 5.

(vii) Event B and C: Getting an odd number on the first die and getting the sum of the numbers on the dice ≤ 5.

(viii) Event A∩ B∩ C: Getting an even number on the first die, getting an odd number on the first die, and getting the sum of the numbers on the dice ≤ 5.

## Question:

A die is thrown describe the following events: (i) A : a number is less than 7 (ii) B : a number is greater than 7 (iii) C : a multiple of 3 (iv) D : a number less than 4 (v) E : an even number greater than 4 (vi) F : a number not less than 3 Also find A∪B,A∩B,B∪C,E∩F,D∩E,A−C,D−E,E∩F′,F′

## Answer:

A: (i) Roll the die. (ii) If the number is less than 7, then event A has occurred.

B: (i) Roll the die. (ii) If the number is greater than 7, then event B has occurred.

C: (i) Roll the die. (ii) If the number is a multiple of 3, then event C has occurred.

D: (i) Roll the die. (ii) If the number is less than 4, then event D has occurred.

E: (i) Roll the die. (ii) If the number is an even number greater than 4, then event E has occurred.

F: (i) Roll the die. (ii) If the number is not less than 3, then event F has occurred.

A∪B = {x|x<7 or x>7} A∩B = ∅ B∪C = {x|x>7 or x is a multiple of 3} E∩F = {x|x>4 and x is an even number} D∩E = {x|x<4 and x is an even number} A−C = {x|x<7 and x is not a multiple of 3} D−E = {x|x<4 and x is an odd number} E∩F′ = {x|x>4 and x is an odd number} F′ = {x|x<3}

## Question:

An experiment involves rolling a pair of dice and recording the numbers that come up describe the following events: A : the sum is greater than 8 B : 2 occurs on either die C : the sum is at least 7 and a multiple of 3 Which pairs of these events are mutually exclusive?

## Answer:

A and B are mutually exclusive because they cannot both occur at the same time. A and C are not mutually exclusive because they can both occur at the same time. B and C are not mutually exclusive because they can both occur at the same time.

## Question:

A die is rolled. Let E be the event “die shows 4” and F be the event “die shows even number”. Are E and F mutually exclusive?

## Answer:

Step 1: Determine what mutually exclusive means. Mutually exclusive means that two events cannot occur at the same time.

Step 2: Analyze the given events. In this case, the event E is “die shows 4” and the event F is “die shows an even number”.

Step 3: Determine if the events are mutually exclusive. No, these events are not mutually exclusive because a die can show both a 4 (an even number) and an even number at the same time.

## Question:

Three coins are tossed once. Let A denote the event ’three heads show", B denote the event “two heads and one tail show” C denote the event “three tails show” and D denote the event ‘a head shows on the first coin" Which events are (i) mutually exclusive? (ii) simple? (iii) compound?

## Answer:

(i) Mutually exclusive events: A and C (ii) Simple events: A, B, C (iii) Compound events: D