What are the odds in favour of getting a spade if a card is drawn from

1.	What are the odds in favour of getting a spade if a card is drawn from a well-shuffled deck of cards? What are the odd in favour of getting a king?
Answer» given: pack of 52 playing cards Formula: P(E) = \(\frac{favorable\ outcomes}{total\ possible\ outcomes}\) one card is drawn from the given pack of cards, total possible outcomes are ⁵²C₁ therefore n(S)=⁵²C₁= 52 (i) let E be the event of getting a spade n(E) = ¹³C₁= 13 P(E) = \(\frac{n(E)}{n(S)}\) P(E) = \(\frac{13}{52}=\frac{1}{4}\) Therefore, probability of event E’ is P(E’) =1 - P(E) P(E') = \(1-\frac{1}{4}=\frac{3}{4}\) Odds in favour of getting a spade is P(E):P(E’) =1:3 (ii) let E be the event of getting a king n(E)= ⁴C₁= 4 P(E) = \(\frac{n(E)}{n(S)}\) P(E) = \(\frac{4}{52}=\frac{1}{13}\) Therefore, probability of event E’ is P(E’) =1 - P(E) P(E') = \(\frac{1}{13}=\frac{12}{13}\) Odds in favour of getting a king is P(E):P(E’) =1:12

What are the odds in favour of getting a spade if a card is drawn from a well-shuffled deck of cards? What are the odd in favour of getting a king?

Answer»

given: pack of 52 playing cards

Formula: P(E) = \(\frac{favorable\ outcomes}{total\ possible\ outcomes}\)

one card is drawn from the given pack of cards, total possible outcomes are ⁵²C₁ therefore

n(S)=⁵²C₁= 52

(i) let E be the event of getting a spade n(E) = ¹³C₁= 13

P(E) = \(\frac{n(E)}{n(S)}\)

P(E) = \(\frac{13}{52}=\frac{1}{4}\)

Therefore, probability of event E’ is

P(E’) =1 - P(E)

P(E') = \(1-\frac{1}{4}=\frac{3}{4}\)

Odds in favour of getting a spade is P(E):P(E’) =1:3

(ii) let E be the event of getting a king

n(E)= ⁴C₁= 4

P(E) = \(\frac{n(E)}{n(S)}\)

P(E) = \(\frac{4}{52}=\frac{1}{13}\)

Therefore, probability of event E’ is

P(E’) =1 - P(E)

P(E') = \(\frac{1}{13}=\frac{12}{13}\)

Odds in favour of getting a king is P(E):P(E’) =1:12

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