Question

In: Statistics and Probability

Question 2. Draw three cards without replacement from a deck of cards and let X be...

Question 2. Draw three cards without replacement from a deck of cards and let X be the number of spades drawn. Sketch the pmf of X and compute E(X).

Question 3. A fair coin is flipped n times. What is the probability of getting a total of k heads if

a) The first flip shows heads

b) The first flip shows tails

c) At least one flip shows heads

Solutions

Expert Solution

We would be looking at question 2 here:

There are 3 cards drawn here without replacement, therefore the probabilities for different values of X is computed here as:

Note that there are 13 spades and 52 - 13 = 39 non spade cards in the deck

P(X = 0) = Number of ways to select 3 cards from 39 non spade cards / Total ways to draw 3 cards from 52 cards = (39c3) / (52c3) = 0.4135

Similarly, P(X = 1) = Number of ways to select 1 card from 13 spade cards * Number of ways to select 2 cards from 39 non spade cards / Total ways to select 3 cards from 52 cards = (13c1)(39c2) / 52c3
= 9633 / 22100
= 0.4359

P(X = 2) = Number of ways to select 2 cards from 13 spade cards * Number of ways to select 1 card from 39 non spade cards / Total ways to select 3 cards from 52 cards = (13c2)(39c1) / 52c3
= 3042 / 22100
= 0.1376

P(X = 3) = Number of ways to draw 3 spade cards from 13 spade cards / Total ways to draw 3 cards from 52 cards = (13c3) / (52c3) = 0.0129

Now using the above probabilities, the expected value of X here is computed as:

E(X) = xP(X = x) = 0*0.4135 + 1*0.4359 + 2*0.1376 + 3*0.0129 = 0.7498

Therefore 0.7498 is the required expected value of X here.

The above PMF is plotted here as:


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