Question

Three cards are drawn from a deck of thirty playing cards separated into three

equally portioned suits: emerald (E), ruby (R), and quartz (Q). Let S be the sample space of

the suits of all drawn hands of three cards.

a. Assuming cards are drawn without replacement, calculate the probability that a quartz is drawn first and a ruby last.

b. Assuming cards are drawn with replacement, calculate the probability of at least two emeralds being drawn

Step 1

**Concept of probability:**

Probability deals with the likelihood of occurrence of a given event. The probability value lies between 0 and 1. An event with probability 1 is considered as certain event and an event with probability 0 is considered as an impossible event. The probability of 0.5 infers of having equal odds of occurring and not occurring of an event.

The general formula to obtain probability of an event A is,

*P*(*A*) = (number of favourable elements for event *A*)/(Total number of elements in the sample space).

The basic properties of probability are given below:

Step 2

**a.**

**Find the probability that quartz (Q) is drawn first and a ruby (R) last:**

The total number of cards in the deck is 30.

The number of quartz (Q) cards is 10.

The number of emerald (E) cards is 10.

The number of ruby (R) cards is 10.

The number of cards drawn is 3.

Here, the 3 cards are drawn in a sequence one after another from the deck without replacement.

Now, the requirement is that 1^{st} card must be quartz (Q), the 3^{rd} card must be ruby (R) and the 2^{nd} card can be anyone of quartz (Q), emerald (E) or ruby (R).

The process is as follows:

- One possibility is: First card is quartz (Q), second card also quartz (Q), third card is ruby (R).
- Second possibility is: First card is quartz (Q), second card also ruby (R), third card is ruby (R).
- Third possibility is: First card is quartz (Q), second card also emerald (E), third card is ruby (R).
- Here, the process is done without replacement. So, the total number of cards for first draw will be 30 and the total number of cards for the second draw will be 29 and the total number of cards for the third draw will be 28.

The probability that quartz (Q) is drawn first and a ruby (R) last is obtained as **0.1149** from the calculation given below:

Step 3

**b.**

**Find the probability that at least 2 emeralds being drawn:**

The total number of cards in the deck is 30.

The number of emerald (E) cards is 10.

The number of cards other than emerald (quartz, ruby) is 20

The number of cards drawn is 3.

Now, the requirement is that there should be at least 2 emerald cards. That means, there should be 2 emeralds or 3 emeralds.

The process is as follows:

- In case of 2 emerald cards. The player has to select 2 emerald cards from 10 emerald cards ...

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