Question

Specify sample spaces for the random experiments (a)-(e) and give mathematical descriptions of the corresponding events. (a) Experiment: A coin is tossed three times. Event: The result of the second toss is “heads”. (b) Experiment: Three indistinguishable coins are tossed at the same time. Event: At most two of the coins show “heads”. (c) Experiment: A die is rolled until each number has appeared at least once. The outcome is the number of rolls needed. Event: Less than 15 rolls are needed. (d) Experiment: n devices labeled with 1, . . . , n are inspected. It is of interest which devices are working and which are not. Event: The first three devices are defective. (e) Experiment: n devices are inspected. Interest lies in the number of defective devices. Event: Exactly three devices are defective.

Answer 1

Solution:-

(Note :- Ω is used for sample space of an experiment and Ω_E is used for sample space of an event.)

a) Sample Space for given experiment

H = "Head" and T = "Tail" when tossing a coin

Ω = {(H,H,H),(H,H,T),(H,T,H),(H,T,T),(T,H,H),(T,H,T),(T,T,H),(T,T,T)}

Now event says "The result of the second toss is “heads” "... So Now all second element need to be H. Sample space will change to

Ω_E = {(H,H,H),(H,H,T),(T,H,H),(T,H,T)} .  So number of outcomes will reduce to 4 (from 8).

b) Sample Space for given experiment

H = "Head" and T = "Tail" when tossing a coin

Ω = {(H,H,H),(H,H,T),(H,T,H),(H,T,T),(T,H,H),(T,H,T),(T,T,H),(T,T,T)}

Now event says "At most two of the coins show “heads” "... So Now all the outcome with 3 heads will be eliminated. Sample space will change to

Ω_E = {(H,H,T),(H,T,H),(H,T,T),(T,H,H),(T,H,T),(T,T,H),(T,T,T)} .  So number of outcomes will reduce to 7 (from 8).

c) Sample Space for given experiment

outcome of this experiment is number of rolls needed. So it will start from 6 and goes to infinity...

Ω = {6,7,8,9,10,11,12,13,14,15,16,........}

Now event says "Less than 15 rolls are needed "... So Now all the outcome above 15 will be eliminated. Sample space will change to

Ω_E = {6,7,8,9,10,11,12,13,14,15} . So number of outcomes will reduce to 10 (from infinite).

d) Sample Space for given experiment

outcome of this experiment is which devices are working and which are not.

D = "Defective" , W = "Working"

Ω = {(D,D,D,D,....) , (W,D,D,D,D,....),........} (Such 2ⁿ number of outcomes are possible ... Each position has 2 possibilities (either D or W) and total n positions are there. i.e. 2x2x2x2x.....(n times) = 2ⁿ )

Now event says "first three devices are defective "... So Now Sample space will change to

Ω_E = {(D,D,D,D,....) , (D,D,D,W,W,....),........} . So number of outcomes will reduce to 2^(n-3) (from 2ⁿ). {First 3 positions are fixed as D and other (n-3) position has 2 possibilities (either D or W)}

e) Sample Space for given experiment

outcome of this experiment is number of defective devices. (possible number defective devices are 1 to n).

Ω = {1,2,3,4,5,6,7,.....,n}

Now event says "Exactly three devices are defective "... So Now Sample space will change to

Ω_E = {3} ... So this event has only one outcome and hence it will be converted to definite event (not random).