Question

A special deck of cards has ten cards. Four are green, three are blue, and three are red. When a card is picked, its color of it is recorded. An experiment consists of first picking a card and then tossing a coin.

a. List the sample space.

b. Let A be the event that a blue card is picked first, followed by landing a head on the coin toss. Find P(A).

c. Let B be the event that a red or green is picked, followed by landing a head on the coin toss. Are the events A and B mutually exclusive? Explain your answer in one to three complete sentences, including numerical justification.

d. Let C be the event that a red or blue is picked, followed by landing a head on the coin toss. Are the events A and C mutually exclusive? Explain your answer in one to three complete sentences, including numerical justification.

Answer 1

Answer:

(a) S = {GH, GT, BH, BT, RH and RT}

(b) The value of P (A) is 0.15.

(c) A and B mutually exclusive.

(d) A and C are not mutually exclusive.

Step-by-step explanation:

There are 10 cards in a special deck of cards: 4 are green (G), 3 are blue (B), and 3 are red (R).

Also when a card is picked, its color of it is recorded. An experiment consists of first picking a card and then tossing a coin.

(a)

The sample space is:

S = {GH, GT, BH, BT, RH and RT}

(b)

A = a blue card is picked first, followed by landing a head on the coin toss

Compute the probability of event A as follows:

$P(A)=P(B)\times P(H)$

         $=\frac{3}{10}\times\frac{1}{2}\\\\=\frac{3}{20}\\\\=0.15$

Thus, the value of P (A) is 0.15.

(c)

B = a red or green is picked, followed by landing a head on the coin toss.

The result of the coin toss is same for both events A and B.

So, consider the events,

A as a blue card is picked first

B as a red or green is picked

There is no intersection point for the two events.

Thus, events A and B mutually exclusive.

(d)

C = a red or blue is picked, followed by landing a head on the coin toss.

The result of the coin toss is same for both events A and C.

So, consider the events,

A as a blue card is picked first

C as a red or blue is picked

There is an intersection point for the two events.

Thus, events A and C are not mutually exclusive.

Answer 2

Part(a): The sample space can be written as shown below:

$S=\{GH,GT,BH,BT,RH,RT\}$

Part(b): The required probability is $P(A)=0.15$

Part(c): The events A and B are mutually exclusive because of $P( A\ AND\ B ) = 0$ are equal to zero.

Part(d): The events A and C are not mutually exclusive because $P( A\ AND\ C ) = 0$ are not equal to zero.

Samples Space:

A sample space is a collection of a set of possible outcomes of a random experiment. The sample space is represented using the symbol, “S”.

Part(a):

A special deck contains ten cards with colors red, green, and blue when the card is picked its color gets recorded, and after that coin will get tossed.

Then the sample space can be written as shown below:

$S=\{GH,GT,BH,BT,RH,RT\}$

Part(b):

If A is the event that a blue card is picked first followed by landing ahead on the coin toss then the outcome it contains is 3 blue cards and 1 head.

Therefore the $P(A)$ is calculated below:

$P(A):P(B)\timesP(H)\\=\frac{3}{10}\times\frac{1}{2}\\ =0.15$

Part(c):

Mutually exclusive events contain a probability $P( A\ AND\ B ) = 0$ that means there is no common outcome between them.

Here, it can be noticed that events A and B cannot happen at the same time. That means, the researcher cannot pick the same cards together. Either it could be red or green.

Hence, events A and B are mutually exclusive because of $P( A\ AND\ B ) = 0$ are equal to zero.

Part(d):

Mutually exclusive events contain a probability $P( A\ AND\ C ) = 0$  which means there is no common outcome between them.

Here, it can be noticed that events A and C can happen at the same time because event C can contain all outcomes of event A.

Hence, events A and C are not mutually exclusive because $P( A\ AND\ C ) = 0$ are not equal to zero.

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