1. Find the sample space and probability distribution that model flip- ping two

Question

1. Find the sample space and probability distribution that model flip- ping two coins. Describe the event "at least one coin comes up heads" formally and compute its probability. 2. We throw two dice. Determine the probability that they both show different numbers under the condition that the sum of both numbers is even 3. (a) people having the same birthday is at least 9/10. (b) Suppose the 4-digit PINs are randomly distributed. How many peo- ple must be in a room such that the probability that two of them have the same PIN is at least 1/2? (Here "4 digits" means that the PIN cannot start with a 0.) Note: You should pay special attention to the direction of the inequality in the formula(s) used for this question. Determine the integer n such that the probability for two of n

Explanation / Answer

1.

Let H denote heads and T denote tails. XY denotes X in 1 coin and Y in 2nd coin

Sample Space S = {HH,TT,HT,TH}

Probability densities are:

P(HH) = P(TT) = P(HT)=P(TH) =1/2*1/2 = 1/4 (Since the two tosses are independent.

P(atleast one coin comes up heads)

= P( HH or HT or TH) = P(HH) + P(HT) + P(TH) = 1/4 +1/4 +1/4 = 3/4

2.

The sum of the numbers will be even only if both are odd or both are even.

Case A : both are odd

# 3 numbers can be chosen on the 1st dice and 2 numbers can be chosen on the 2nd dice

P(A) = 3*2/(6*6)

Case B: both are even (Same as odd)

# 3 numbers can be chosen on the 1st dice and 2 numbers can be chosen on the 2nd dice

P(B) =3*2/(6*6)

Total probability = 2*6/36 = 1/3

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