Question

Let X be the number of spades that show up when randomly choosing three cards from a standard 52 card deck. (without replacement). Construct a probability distribution for X on the table to the left.

Use fractions for probabilities. The find the mean and standard deviation.

x	P(X=x)

Mean: 2 Decimal Places

Standard Deviation: 2 Decimal Places

Answer 1

Solution:
No. of spades in Standard deck of cards = 13
Total Cards in standard deck = 52
We need to construct a probability distribution for X when we choose three cards from a Standard 52 cards deck.
P(0 Spades) = Total No. of ways to select 3 cards other than spade out of 39 cards/ Total no. of ways to select 3 cards from 52 cards = 39C3 / 52C3 = 9139/22100 = 0.4135
P(1 Sapde) = Total No. of ways to select 3 cards which have 1 spade out of 13 and 2 other than spade out of 39 cards/ Total no. of ways to select 3 cards from 52 cards = 13C1 * 39C2 / 52C3 = 13*741/22100 = 0.4359
P(2 Spade) = Total No. of ways to select 3 cards which have 2 spade out of 13 and 1 other than spade out of 39 cards/ Total no. of ways to select 3 cards from 52 cards = 13C2*39C1/52C3 = 78*39/22100 = 0.1376
P(3 Sapde) = Total No. of ways to select 3 cards which have 3 spade out of 13/ Total no. of ways to select 3 cards from 52 cards = 13C3/52C3 = 286/22100 = 0.0129
So probability distribution can be written as

X	P(X)
0	0.4135
1	0.4359
2	0.1376
3	0.0129

Mean of probability distribution Can be calculated as
Mean = (Xi*P(Xi)) = (0*0.4135) + (1*0.4359) + (2*0.1376) + (3*0.0129) = 0 + 0.4359 + 0.2752 + 0.0387 = 0.7498 or 0.75
Mean = 0.75
Standard deviation of Probability distribution can be calculated as
Standard deviation = Sqrt((Xi-mean)^2 * P(Xi)

X	P(X)	(Xi-mean)	(Xi-mean)^2	(Xi-mean)^2 *P(Xi)
0	0.4135	-0.75	0.5625	0.23259375
1	0.4359	0.25	0.0625	0.02724375
2	0.1376	1.25	1.5625	0.215
3	0.0129	2.25	5.0625	0.06530625

Standard deviation = sqrt(0.23259275 + 0.02724375 + 0.215 + 0.06530625) = sqrt(0.5401) = 0.73