In: Statistics and Probability
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
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: