Probability and counting

• Three cards are dealt from a pack of 52
  • a. find the probability that one club and two hearts are dealt, in any order.

P(octh)
= \frac{13C_{1} *13C_{2}}{52C_{3}}
= \frac{1014}{22100}
=\frac{39}{850}

$$ P(CHH)=\frac{13*13*12}{52*51*50}=\frac{2028}{132600}=\frac{13}{850}

• In a lottery 6 numbers from 1 to 40 are chosen by selecting balls from a barrel without replacement. You win if you choose all 6 numbers. What is the probability that you win?
  • $P(win)=\frac{1}{40C_6}=\frac{1}{388380}$
• A bag contains three red, seven yellow and five blue balls. If three balls are drawn from the bag simultaneously, find the probability that:
  • all three balls are yellow,
    • $P("3y")=\frac{7C_3}{15C_3}=\frac{1}{13}$
  • all the balls are of the same colour,
    • $P(same)=\frac{7C_1\ast 3C_1\ast 5C_1}{15C_3}$