Referring to the AP Inter 1st Year Maths Study Material Chapter 14 Probability Exercise 14d Solutions makes it easier to understand complex problems.
AP Inter 1st Year Maths Probability Solutions Exercise 14d
In each of the following Exercises 1 to 7, describe the sample space for the indicated experiment.
Question 1.
A coin is tossed four times.
Solution:
Let S’ be a Sample space. Each toss has two outcomes: H(Head), T(Tail).
Total outcomes = 24 = 16.
S = {HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT}.
Question 2.
A coin is tossed and a dice is thrown.
Solution:
Let ‘S’ be a Sample space, Coin → {H, 1, Dice → {1, 2, 3, 4, 5, 6)
S = {(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)}
Total outcomes = 12.
Question 3.
A coin is tossed and then a dice is rolled only in case a head is shown on the coin.
Solution:
Let S’ be a Sample space. If Head : roll dice → {1, 2, 3, 4, 5, 6), If Tail : no dice roll → T
S = {T, (H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6)}
Total outcomes = 7
Question 4.
2 boys and 2 girls are in Room X, and 1 boy and 3 girls in Room Y. Specify the sample space for the experiment in which a room is selected and then a person.
Solution:
Let ‘S’ be a Sample Space. Room X : B1, B2, G1, G2 Room Y : B3, G4, G3, G4
S = {(X, B1), (X, B2), (X, G1), (X, G2), (Y, B3), (Y, B4), (Y, G3), (Y, G4)}
Total outcomes = 8
Question 5.
One dice of red colour, one of white colour and one of blue colour are placed in a bag. One dice is selected at random and rolled, its colour and the number on its uppermost face is noted. Describe the sample space.
Solution:
Dice colour → {R, W, B}, Each with outcomes 1 to 6.
S = {(R, 1), (R, 2), …, (R, 6), (W, 1), … , (W, 6), (B, 1) … , (B, 6)}
Total outcomes = 3 × 6 = 18
Question 6.
An experiment consists of recording boy-girl composition of families with 2 children.
i) What is the sample space if we are interested in knowing whether it is a boy or girl in the order of their births?
ii) What is the sample space if we are interested in the number of girls in the family?
Solution:
i) Order matters [first born, second bom) possible combinations
S = {BB, BG, GB, GG} (B = Boy, G = Girl)
ii) Only number of girls matters
S = {0, 1, 2} (0 girls → BB, 1 Girl → BG (or) GB, 2 girls → GG).
Question 7.
A box contains 1 red and 3 identical white balls. Two balls are drawn at random in succession without replacement. Write the sample space for this experiment.
Solution:
Let R = Red, W = White, Balls : {R, W1, W2, W3} {W’s are identical, so indistinct)
Unique Combinations: RW, WR, WW
Hence, S = {RW, WR, WW}
Note : W1, W2, W3 are identical, so (W1, W2) – (W2, W3) = WW.
Question 8.
An experiment consists of tossing a coin and then throwing it second time if a head occurs. If a tail occurs on the first toss, then a dice is rolled once. Find the sample space.
Solution:
Toss a coin. If head, toss again. If tail, roll a dice.
Sample Space, S :
First toss = H → toss again : (H, H), (H, T).
First toss = T → roll dice : (T, 1), (T, 2), … (T, 6).
S = {(H, H), (H, T), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)}
Total outcomes = 2(From HH, HT) + 6 (From T1 – T6) = 8
Question 9.
Suppose 3 bulbs are selected at random from a lot. Each bulb is tested and classified as defective (D) or non-defective (N). Write the sample space of this experiment.
Solution:
3 bulbs are to be selected at random from the lot. Each bulb in the lot is tested and classified as defective (D) or non-defective (N). The sample space of this experiment is given by
S = {DDD, DDN, DND, NDN, NND, NNN)}.
Question 10.
A coin is tossed. If the outcome is a head, a dice is thrown. If the dice shows up an even number, the dice is thrown again. What is the sample space for the experiment?
Solution:
When a coin is tossed, the possible outcomes are head (H) and tail (T).
When a dice is thrown, the possible outcomes are 1, 2, 3, 4, 5 or 6 S = {T, H1, H3, H5, H21, H22, H23, H24, H25, H26, H41, H42, H43, H44, H45, H46, H61, H62, H63, H64, H65, H66}.
Question 11.
The numbers 1, 2, 3 and 4 are written separately on four slips of paper. The slips are put in a box and mixed thoroughly. A person draws two slips from the box, one after the other, without replacement. Describe the sample space for the experiment.
Solution:
Let ‘S’ be a Sample space.
Permutations of 2 slips out of 4 :
S = {(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2), (3, 4), (4, 1), (4, 2), (4, 3)}
Total outcomes = 4P2 = 12
Question 12.
An experiment consists of rolling a dice and then tossing a coin once if the number on the dice is even. If the number on the dice is odd, the coin is tossed twice. Write the sample space for this experiment.
Solution:
Experiment : Roll a dice, if it shows an even number, toss a coin once. If odd, toss the coin twice.
The possible outcomes when rolling dice are {1, 2, 3, 4, 5, 6}.
Outcomes for even numbers on dice : (2, H), (2, T), (4, H), (4, T), (6, H), (6, T).
Outcomes for odd numbers on dice : (1, HH), (1, TT), (1, TH), (1, HT), (3, HH), (3, TT), (3, TH), (3, HT), (5, HH), (5, TT), (5, TH), (5, HT).
So, S = {(2, H), (2, T), (4, H), (4, T), (6, H), (6, T), (1, HH), (1, TT), (1, TH), (1, HT), (3, HH), (3, TT), {3, TH), (3, HT), (5, HH), (5, TT), (5, TH), (5, HT)}
Total outcomes = 6 (even) + 12 (odd with coin twice) = 18
S = {(6), (x1, 6), (x1, x2, 6), …} where each xi ∈ {1, 2, 3, 4, 5}.
This is an infinite sample space of sequences ending with 6 and prior values being from (1, 2, 3, 4, 5}.
Question 13.
A coin is tossed. If it shows a tail, we draw a ball from a box which contains 2 red and 3 black balls. If it shows head, we throw a dice. Find the sample space for this experiment.
Solution:
Experiment : A coin is tossed if tail, draw a ball from box with 2 red and 3 black balls. If head, throw a dice.
Sample space, S :
If coin is Head → roll dice : H1, H2 … , H6
If coin is tail draw ball : TR1, TR2 (red), TB1, TB2, TB3 (Black).
5 = (H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, Red 1), (T, Red 2), (T, Black 1), (T, Black 2), (T, Black 3).
Total outcomes = 6 + 5 = 11
Question 14.
A dice is thrown repeatedly untill a six comes up. What is the sample space for this experiment ?
Solution:
Experiment: A dice is thrown repeatedly untill a six comes up.
Sample space, S :
Stop only when 6 comes possible outcomes :
6
1, 6
1, 1, 6
2, 4, 3, 6
……………..
S = {6, (1, 6) ………….. (1, 1, 6) ………… (2,4, 3, 6) …………}
All sequences ending with 6 and before that only number (or) numbers from 1 to 5.