Question

I will invite four friends to a party. Four invitations are prepared along with named envelopes−one for each friend. The invitations are put into the envelopes at random.

(a) Describe a sample space S for this experiment. Give an example of one outcome (sample point). How many outcomes are in your sample space?

(b) Let Y denote the number of invitations that are put into the correctly named envelope. Find the probability mass function of Y .

Answer 1

Here lets say four friends are 1,2,3 & 4 and the named envelopes are A,B,C and D respectively.

so Herethe sample space S for this experimet is that for any envelope there is four possiilities of invitations to be put in that.

so there are 4 possibilities of invitation for envelope A, 3 for envelope B, 2 for envelope C and 1 for envelope D.

so here

Example of one sample point is = { 1A,2C, 3D, 4B}

like that there are 24 combinations to be there

(b) Here Y is the pmf of number of invitations that are put into the correctly named envelope.

so, Y would be four only one time when all inviations are put into correct enevelipe like {1A,2B,3C,4D{

so p(Y = 4) = 1/24

now, if Y = 3 then automatically, Y would be 4 so

p(Y = 3) = 0

Now for Y = 2, we know that 2 are in correct place so total combination for that = ⁴C₂ = 6 and rest of two can wrong only at one position when they are in different envelope.

so,

p(Y = 2) = 6/24= 0.25

Now, for Y = 1, Any one be in correct envelope so there are four such possivilities. Rest 3 shall be in wrong envelope. So, there are total 6 possibilities for these 3 invitations and only one is when they are in right place.

so,

p(y = 1) = 4 * 2/24= 8/24 = 1.3

So, now

p(y = 0) = = 1 - (1/24 + 6/24 + 8/24) = 9/24 = 3/8

so,

p(y) = 9/24 ; y = 0

= 8/24; y = 1

= 6/24 ; y = 2

= 0; y = 3

= 1/24 ; y = 4