Suppose that a coin is tossed three times and the side showing face up on each toss is noted. Suppose also that on each toss heads and tails are equally likely. Let HHT indicate the outcome heads on the first two tosses and tails on the third, THT the outcome tails on the first and third tosses and heads on the second, and so forth.

(a) Using set-roster notation, list the eight elements in the sample space whose outcomes are all the possible head-tail sequences obtained in the three tosses.
(b) Write each of the following events as a set, in set-roster notation, and find its probability.(i)The event that exactly one toss results in a head.

Suppose that a coin is tossed three times and the side showing face up on each toss is noted. Suppose also that on each toss heads and tails are equally likely. Let HHT indicate the outcome heads on the first two tosses and tails on the third, THT the outcome tails on the first and third tosses and heads on the second, and so forth.

(a) Using set-roster notation, list the eight elements in the sample space whose outcomes are all the possible head-tail sequences obtained in the three tosses.

(b) Write each of the following events as a set, in set-roster notation, and find its probability.

(i)The event that exactly one toss results in a head.

(ii) The event that at least two tosses result in a head

(iii) The event that no head is obtained.

Answer:

a) SS = {HHH, THH, HTH, HHT, THT, TTH, HTT, TTT}

i) P(1 head) = 0.375

ii) P(at least 2 heads) = 0.5

iii) P(no heads) = 0.125

Step-by-step explanation:

a) A coin is tossed three times and the side showing face up on each toss is noted.

The total number of possible outcome are 2³ = 8

The sample space is given by

SS = {HHH, THH, HTH, HHT, THT, TTH, HTT, TTT}

b) Write each of the following events as a set, in set-roster notation, and find its probability.

(i)The event that exactly one toss results in a head.

In this case we need to include only those outcomes where we have exactly on head,

E(1 head) = {THT, TTH, HTT}

So there are 3 such outcomes, the probability is

P(1 head) = no. of desired outcomes/total number of outcomes

P(1 head) = 3/8

P(1 head) = 0.375

(ii) The event that at least two tosses result in a head

In this case we need to include only those outcomes where we have at least two heads which means two or greater than two,

E(at least 2 heads) = {HHH, THH, HTH, HHT}

So there are 4 such outcomes, the probability is

P(at least 2 heads) = 4/8

P(at least 2 heads) = 0.5

(iii) The event that no head is obtained.

In this case we need to include only those outcomes where we have no heads at all.

E(no heads) = {TTT}

So there is only 1 such outcome, the probability is

P(no heads) = 1/8

P(no heads) = 0.125