Answer to Question #169832 in Statistics and Probability for Arche

Question

1.Three coins are tossed. Let T be the number of tails that occurs. Determine the values of the random variable T.

2. A coin is flipped four times. Let T be the number of tails that come out. Determine the values of random variable T.

3. Two balanced dice are rolled. Let S be the random variable denoting the sum of the number of dots that will appear. Determine the values of the random variable S.

4. Let X be the number of boys in a family of four children. Determine the values of the random variable X.

Answer 1

1.Three coins are tossed:

For each toss, the outcome belongs to one of the following: {H, T}

where H is heads and T is tails.

We want to count the total number of tails obtained from those 3 tosses. 

Observe that any one of the following cases may happen:

1. {TTT} - All the three outcomes are 'Tail'. Hence, T = 3 in this case.
2. e.g. {TTH} - Any 2 of the 3 outcomes are 'Tail' and the remaining one is a 'Head' Hence, T = 2 in this case.
3. e.g. {THH} - Any 1 of the 3 outcomes are 'Tail' and the remaining two are 'Head' Hence, T = 1 in this case.
4. {HHH} - All the three outcomes are 'Head'. So the number of tails is 0. Hence, T = 0 in this case.

Note that, the 4 cases listed above explores all possible outcomes. Hence, the random variable T takes any one value from {0, 1, 2, 3}.

2. A coin is flipped four times:

For each toss, the outcome belongs to one of the following: {H, T}

where H is heads and T is tails.

We want to count the total number of tails obtained from those 4 tosses. 

Observe that any one of the following cases may happen:

1. {TTTT} - All the four outcomes are 'Tail'. Hence, T = 4 in this case.
2. e.g. {TTTH} - Any 3 of the 4 outcomes are 'Tail' and the remaining one is a 'Head' Hence, T = 3 in this case.
3. e.g. {TTHH} - Any 2 of the 4 outcomes are 'Tail' and the remaining two are 'Head' Hence, T = 2 in this case.
4. e.g. {THHH} - Any 1 of the 4 outcomes are 'Tail' and the remaining three are 'Head' Hence, T = 1 in this case.
5. {HHHH} - All the four outcomes are 'Head'. So the number of tails is 0. Hence, T = 0 in this case.

Note that, the 5 cases listed above explores all possible outcomes. Hence, the random variable T takes any one value from {0, 1, 2, 3, 4}.

3. Two balanced dice are rolled:

For each rolled dice, the outcome belongs to one of the following: {1, 2, 3, 4, 5, 6}

Thus we can define the following sample space for a two balanced dice:

(1,1) (1,2) (1,3) (1.4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

Therefore the sum of the number of dots that will appear S takes any one value from {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

4. Let X be the number of boys in a family of four children:

Observe that any one of the following cases may happen:

1. {BBBB} - All the four outcomes are 'Boy'. Hence, X = 4 in this case.
2. e.g. {BBBG} - Any 3 of the 4 outcomes are 'Boy' and the remaining one is a 'Girl' Hence, X = 3 in this case.
3. e.g. {BBGG} - Any 2 of the 4 outcomes are 'Boy' and the remaining two are 'Girl' Hence, X = 2 in this case.
4. e.g. {BGGG} - Any 1 of the 4 outcomes are 'Boy' and the remaining three are 'Girl' Hence, X = 1 in this case.
5. {GGGG} - All the four outcomes are 'Girl'. So the number of boys is 0. Hence, X = 0 in this case.

Note that, the 5 cases listed above explores all possible outcomes. Hence, the random variable X takes any one value from {0, 1, 2, 3, 4}.