Basic ProbabilitySection 7.1

Definitions• Sample Space:• The set of all possible outcomes in a given experiment or

situation• Often denoted by S

• Event:• A subset of outcomes from a sample space.• Usually denoted by E

• Probability:• The quotient:• Usually denoted by p(E) or just p.• Note: 0 ≤ p(E) ≤ 1

Examples 1• 1) What is the probability of choosing a blue ball from a bag

containing 4 blue and 5 black balls?

• |S| = 9, |E| = 4, so p(E) = 4/9 = .44444…

• 2) What is the probability of rolling a 7 with a pair of fair dice?

• |S| = 6 x 6 = 36, E = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}

p(E) = 6/36 = 1/6 = .16666666…

Examples 2• 3) What is the probability of guessing a 4-digit PIN?

• |S| = 10 x 10 x 10 x 10 = 10000, |E| = 1

p(E) = 1/10000 = .0001

• 4) What is the probability of winning a Pick-Six Lottery chosen from 40 numbers?

• |S| = C(40,6) = 40!/(34!6!) = 3,838,380

p(E) = 1/|S| = .0000002605265…

Examples 3• 5) What is the possibility of getting dealt a Royal Flush in 5-

card poker?

• |S| = C(52,5) = 52!/(5!47!) = 2,598,960

|E| = 4

p(E) = .000001539…

Examples 4• 6) What is the possibility of getting dealt a Full House?

• First consider getting 3 Queens and 2 Aces• This is different than 3 Aces and 2 Queens (order matters!)• How many ways can you get 3 Queens? C(4,3) = 4

How many ways can you get 2 Aces? C(4,2) = 6p(3Q2A) = (4)(6)/2,598,360

• How many ways can you choose two kinds, where order matters?• P(13,2) = (13)(12)

• So p(E) = (13)(12)(4)(6)/2,598,960 = .0014…

Probabilities of Complements of Events• A complement of an event E, denoted by or E', is the

complement of the set E relative to S.

• Therefore, |E| + |E'| = |S|• ⇒ p(|E|) = 1 – P(E'), and vice-versa

• 7) What is the probability that a sequence of 10 random bits contains a 0-bit?

p(E) = 1 – p(E') = 1 – 2-10 = 1023/1024

Probabilities of Unions• Remember that |A B| = |A| + |B| - |A ∩ B|∪

• Therefore, if A and B are events in S, then

p(A B) = |A|/|S| + |B|/|S| - |A ∩ B|/|S|∪

= p(A) + p(B) – p(|A ∩ B|)

• 8) What is the probability of selecting a number from [1,100] that is divisible by 2 or 5?

• p(E) = 50/100 + 20/100 – 10/100 = 3/5 = .6