Playing with a deck of 54 cards including Jokers which act a

Playing with a deck of 54 cards (including Jokers which act as a wild card that represents any card). What is the probabilty that a five card poker hand has a four of a kind?

Solution

Total number of ways in which 5 cards can be drawn out of 54 = ⁵⁴C₅ = 3162510

Number of ways to get four of a kind = (¹³C₁ * ⁴C₃ ) * ( ²C₁) *( ⁴⁸C₁) + (¹³C₁ * ⁴C₂ ) * ( ²C₂) *( ⁴⁸C₁)

+(¹³C₁ * ⁴C₄ ) * ( ²C₀) *( ⁴⁸C₁)

= 4992 + 3744 + 624

= 9360

Probabilty that a five card poker hand has a four of a kind = Number of ways to get four of a kind / Total number of ways in which 5 cards can be drawn out of 54

= 9360 / 3162510

= 0.00296