STATISTICS ASSIGNMENT 1

  

STA 111

1. THE ANSWERS

a) The probability of drawing one ace from a deck of 52 cards is 4/52, since there are four aces in the deck. The probability of drawing a second ace, assuming the first ace was not replaced (without replacement), is 3/51. Therefore, the probability of drawing two aces is:

(4/52) x (3/51) = 12/2652

Simplifying this fraction, we get:

12/2652 = 1/221

Therefore, the chance of drawing two aces is 1/221.

b) Note: a pack of card (52), contains 4 kings, 4 queens and 4 knaves.

The number of ways to choose one king from four kings is ⁴C₁ = 4. Similarly, the number of ways to choose one queen from four queens is ⁴C₁ = 4, and the number of ways to choose one knave from four knaves is also ⁴C₁ = 4. Therefore, the total number of ways to choose one king, one queen, and one knave is:

4 x 4 x 4 = 64

The total number of ways to choose three cards from a deck of 52 cards is ⁵²C₃ = 22,100.

Therefore, the probability of choosing a king, a queen, and a knave is:

64/22,100

Simplifying this fraction, we get:

64/22,100 = 16/5,525

Therefore, the chance of drawing a king, a queen, and a knave is 16/5,525.

 

2. THE ANSWERS

i) Note: they are 13 diamond card from a pack of card (52).

The number of ways to choose 4 cards from a deck of 52 cards is  = 270,725. The number of ways to choose 4 diamonds from the 13 diamonds in the deck is ¹³C₄ = 715. Therefore, the probability of choosing 4 diamonds is:

715/270,725

Simplifying this fraction, we get:

715/270,725 = 31/11,725

Therefore, the probability that all 4 cards are diamonds is 31/11,725.

ii) Note they are 13 hearts and 13 spades cards in a pack of card:

The number of ways to choose 2 spades from the 13 spades in the deck is ¹³C₂ = 78. The number of ways to choose 2 hearts from the 13 hearts in the deck is also ¹³C₂ = 78. Therefore, the total number of ways to choose 2 spades and 2 hearts is:

78 x 78 = 6,084

The total number of ways to choose 4 cards from a deck of 52 cards is ⁵²C₄ = 270,725.

Therefore, the probability of choosing 2 spades and 2 hearts is:

6,084/270,725

Simplifying this fraction, we get:

6,084/270,725 = 12/535

Therefore, the probability that 2 cards are spades and 2 cards are hearts is 12/535.

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