GSEB Class 11 Maths Solutions Chapter 16 Probability Exercise 16.1

Question 1. A coin is tossed three times.
Answer: When one coin is tossed three times, the possible results of the experiment are HHH, HHT, HTH, THH, TTH, THT, HTT, TTT. The complete sample space is therefore given by S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}.
In simple words: When you flip a coin three times, you list all the possible sequences of heads (H) and tails (T) that can happen.

Exam Tip: For coin toss experiments, remember that each toss is independent. For 'n' tosses, the total number of outcomes is \( 2^n \). List them systematically to avoid missing any.

Question 2. A die is thrown two times.
Answer: When a die is thrown two times, the complete sample space S for this experiment is given by S = {(1, 1), (1, 2), ..., (1, 6), (2, 1), (2, 2), ..., (2, 6), ..., (6, 1), ..., (6, 6)}. This includes all possible combinations where each roll can be any number from 1 to 6.
In simple words: When a die is rolled twice, the sample space includes every pair of numbers you could get, like (1,1), (1,2), all the way up to (6,6).

Exam Tip: For rolling a die 'n' times, the total number of outcomes is \( 6^n \). When listing, it's often helpful to organize by the first roll, then the second, and so on.

Question 3. A coin is tossed four times.
Answer: When a coin is tossed four times, the full sample space S linked with this experiment is S = {HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT}. This means there are \( 2^4 = 16 \) different possible sequences of heads and tails.
In simple words: When a coin is flipped four times, you list all 16 possible combinations of heads and tails that could show up.

Exam Tip: When listing many outcomes, maintain a pattern (e.g., start with all H's, then one T at the end, then one T moving left, etc.) to ensure all possibilities are covered and none are repeated.

Question 4. A coin is tossed and a die is thrown.
Answer: When a coin is tossed and a die is thrown, the complete sample space S associated with the experiment is S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}. This includes all six possible die outcomes for both a head and a tail.
In simple words: When you flip a coin and roll a die, the possible results are a head with any number from 1 to 6, or a tail with any number from 1 to 6.

Exam Tip: For combined experiments, multiply the number of outcomes for each individual experiment to find the total sample space size. Here, 2 (coin) * 6 (die) = 12 outcomes.

Question 5. A coin is tossed and then a die is rolled only in case a head is shown on the coin.
Answer: When a coin is tossed and a die is rolled only if a head shows on the coin, the sample space S linked with the experiment is S = {H1, H2, H3, H4, H5, H6, T}. If a tail appears, no die is rolled, so 'T' is a single outcome.
In simple words: If you get a head, you then roll a die and record H with the number. If you get a tail, you just record T and stop.

Exam Tip: Pay close attention to conditional statements ("only in case", "if... then..."). These limit the experiment's path, affecting the final sample space.

Question 6. 2 boys and 2 girls are in Room X and 1 boy and 3 girls are in Room Y. Specify the sample space for the experiment in which a room is selected and then a person.
Answer: Let's denote the boys in Room X as B1, B2 and the girls as G1, G2. For Room Y, let the boy be B3 and the girls be G3, G4, G5. The sample space S for the experiment where a room is first chosen and then a person is selected is S = {XB1, XB2, XG1, XG2, YB3, YG4, YG5}. This lists every possible room-person combination.
In simple words: First, you choose either Room X or Room Y. Then, you pick one person from that chosen room. The sample space lists all the people you could pick from each room.

Exam Tip: Clearly define your labels for components (like B1, G1 for individuals) to keep the sample space organized and understandable, especially in complex scenarios.

Question 7. One die of red colour, one white and one of blue colour are placed in a bag. One die is selected at random and rolled, its colour and the number on its uppermost face is noted. Describe the sample space.
Answer: Let R, W, and B represent the red, white, and blue dice respectively. The complete sample space S for selecting one die at random and then rolling it is S = {R1, R2, R3, R4, R5, R6, W1, W2, W3, W4, W5, W6, B1, B2, B3, B4, B5, B6}. This includes all possible color-number pairs.
In simple words: You pick a die (red, white, or blue) and then roll it. The sample space shows the color of the die you picked and the number you rolled on it.

Exam Tip: For experiments with multiple stages, consider all possible paths. Here, each color choice leads to 6 more outcomes, so it's 3 (colors) * 6 (die faces) = 18 total outcomes.

Question 8. An experiment consists of recording boy-girl composition of families with 2 children.
1. What is the sample space, if we are interested in knowing whether it is a boy or girl in the order of their births?
2. What is the sample space, if we are interested in the number of girls in a family?
Answer:
1. The sample space S, when we are interested in the order of birth (boy or girl) for families with two children, is given by S = {BB, BG, GB, GG}. Each letter represents the gender of a child, and the order matters.
2. The sample space S, when we are interested in the number of girls in a family of two children, is given by S = {0, 1, 2}. This means a family can have zero, one, or two girls.
In simple words: For two children, if you care about the birth order, list all boy/girl combinations like Boy-Boy. If you only care about how many girls there are, then it can be 0, 1, or 2 girls.

Exam Tip: Always read the question carefully to understand what specific aspect of the experiment the sample space should describe. This determines whether order matters or if you're counting a particular event.

Question 9. 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.
Answer: There is 1 red ball and 3 identical white balls in a box. The sample space S for selecting two balls at random in succession without replacement is S = {RW, WR, WW}. Since the white balls are identical, drawing a specific white ball first or second doesn't change the outcome 'W', but the order of R and W still matters.
In simple words: You have one red ball and three white balls. You pick two balls, one after the other, and don't put the first one back. The possible picks are Red then White, White then Red, or White then White.

Exam Tip: "Without replacement" means the pool of items changes after the first draw. "In succession" implies order matters. When items are identical, treat them as such in the final sample space, but consider distinct paths during analysis.

Question 10. An experiment consists of tossing a coin and then tossing it second time, if a head occurs. If a tail occurs on the first toss, then a die is tossed once. Find the sample space.
Answer: The sample space S for this experiment is given by S = {HH, HT, T1, T2, T3, T4, T5, T6}. If the first toss is a head, the coin is tossed again (HH, HT). If the first toss is a tail, a die is tossed once (T1, T2, T3, T4, T5, T6).
In simple words: If the first coin flip is heads, you flip the coin again. If it's tails, you roll a die. The sample space lists all the possible results from these steps.

Exam Tip: For conditional experiments, it can be helpful to draw a tree diagram to visualize the different paths and their corresponding outcomes in the sample space.

Question 11. 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.
Answer: The sample space S for selecting three bulbs randomly from a lot and classifying each as defective (D) or non-defective (N) is S = {DDD, DDN, DND, DNN, NDD, NDN, NND, NNN}. Each sequence represents the classification of the three bulbs in the order they were selected.
In simple words: When you pick three light bulbs and check if each is broken (D) or not (N), the sample space lists all the possible combinations of working and broken bulbs.

Exam Tip: Similar to coin tosses, if there are 'k' categories and 'n' items, the number of outcomes is \( k^n \). For 3 bulbs and 2 categories (D/N), it's \( 2^3 = 8 \) outcomes.

Question 12. A coin is tossed. If the outcome is a head, a die is thrown. If the die shows up an even number, the die is thrown again. What is the sample space for this experiment?
Answer: The sample space S for this experiment is S = {T, H1, H3, H5, H21, H22, H23, H24, H25, H26, H41, H42, H43, H44, H45, H46, H61, H62, H63, H64, H65, H66}. If a tail (T) appears, the experiment ends. If a head (H) appears, a die is thrown. If that die roll is odd (1, 3, 5), the experiment ends. If the die roll is even (2, 4, 6), the die is thrown again. So, H21 means Head, then 2, then 1, and so on. (Correcting H46 to H45 based on standard die sequence, and including H45, H64 which were missing from the OCR output based on the rule).
In simple words: You flip a coin. If it's tails, you stop. If it's heads, you roll a die. If the die is odd, you stop. If the die is even, you roll it a second time. The sample space lists all the different ways these steps can happen.

Exam Tip: Complex conditional experiments require a methodical approach. Draw a decision tree to map all possible sequences of events and outcomes to ensure completeness and accuracy.

Question 13. The number 1, 2, 3 and 4 are written separately on four slips of paper. The slips are then 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.
Answer: Four slips, numbered 1, 2, 3, and 4, are in a box. Two slips are drawn from it one after the other without putting the first one back. The sample space S for this experiment is given by 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)}. This lists all ordered pairs where the numbers are distinct.
In simple words: You pick two numbered slips from a box, one by one, without putting the first one back. The sample space shows all the different pairs of numbers you could pick in order.

Exam Tip: "Without replacement" means the available options decrease after each draw. "One after the other" indicates that the order of drawing matters, making (1,2) different from (2,1).

Question 14. An experiment consists of rolling a die and then tossing a coin once, if the number on the die is even. If the number on the die is odd, the coin is tossed twice. Write the sample space for this experiment.
Answer: An experiment involves rolling a die. If the number on the die is even, a coin is tossed once. If the number on the die is odd, the coin is tossed twice. The sample space S for this experiment is given by S = {1HH, 1HT, 1TH, 1TT, 2H, 2T, 3HH, 3HT, 3TH, 3TT, 4H, 4T, 5HH, 5HT, 5TH, 5TT, 6H, 6T}. This combines the die roll with the subsequent coin tosses.
In simple words: Roll a die first. If it's an even number (2,4,6), flip a coin once. If it's an odd number (1,3,5), flip a coin twice. The sample space lists all possible outcomes.

Exam Tip: Break down the experiment into its conditional stages. For odd die rolls (1, 3, 5), there are \( 2^2 = 4 \) coin outcomes each. For even die rolls (2, 4, 6), there are \( 2^1 = 2 \) coin outcomes each.

Question 15. 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 die. Find the sample space for this experiment.
Answer: An experiment consists of tossing a coin. If it shows a tail, a ball is drawn from a box containing 2 red (R) and 3 black (B) balls. If it shows a head, a die is thrown. The sample space for this experiment is S = {TR1, TR2, TB1, TB2, TB3, H1, H2, H3, H4, H5, H6}. This lists outcomes from drawing balls after a tail, and rolling a die after a head.
In simple words: If you flip a tail, you pick a ball from a box (two red, three black). If you flip a head, you roll a die. The sample space shows all the possibilities like Tail-Red ball 1, or Head-Number 4.

Exam Tip: Distinguish between events that trigger different subsequent experiments. The initial coin toss determines which secondary experiment takes place, leading to different branches in the sample space.

Question 16. A die is thrown repeatedly until 6 comes up. What is the sample space for the experiment?
Answer: The experiment involves throwing a die repeatedly until the number 6 appears. The sample space S for this experiment consists of sequences where the last outcome is 6, and all preceding outcomes are not 6. Since there are five non-6 outcomes (1, 2, 3, 4, 5) for each throw before a 6, the sample space is: S = {6, (1,6), (2,6), (3,6), (4,6), (5,6), (1,1,6), (1,2,6), (1,3,6), (1,4,6), (1,5,6), (2,1,6), ..., (5,5,6), (1,1,1,6), ...}. This infinite sample space captures all possible sequences ending with a 6.
In simple words: You keep rolling a die until you get a 6. The sample space lists all the ways this can happen, like rolling a 6 on the first try, or rolling a 1 then a 6, or rolling a 1 then a 1 then a 6, and so on forever.

Exam Tip: When an experiment involves repetition until a specific event occurs, the sample space can be infinite. List the initial possible outcomes and use ellipses (...) to indicate the continuing pattern.