Theory of Probability Ex 16.1 Class 11 NCERT: Sample Spaces

Fundamental Concepts for Exercise 16.1

Random Experiment

An experiment whose outcome cannot be predicted with certainty, but all possible outcomes are known. For example, tossing a coin or rolling a die. It must have more than one possible outcome and be repeatable under similar conditions.

Outcome

A possible result of a random experiment. For example, getting 'Heads' is one outcome of tossing a coin. '4' is one outcome of rolling a die.

Sample Space (S)

The set of all possible outcomes of a random experiment. It is denoted by the capital letter S. For a coin toss, S = {H, T}. For a die roll, S = {1, 2, 3, 4, 5, 6}.

How to Describe a Sample Space

The main task in Exercise 16.1 is to correctly identify and write down the sample space for a given random experiment. The key is to be systematic and thorough. Don't leave any possibilities out! Think about the experiment in stages. If you toss two coins, think about the outcome of the first toss, and then the outcome of the second toss for each result of the first. A tree diagram can be very helpful. For example, the first coin can be H or T. If it's H, the second can be H or T (giving HH, HT). If the first is T, the second can be H or T (giving TH, TT). This systematic approach ensures you list all outcomes: {HH, HT, TH, TT}. The same logic applies to more complex experiments, like rolling a die and then tossing a coin. For each of the 6 outcomes of the die, there are 2 outcomes for the coin. So, the total number of outcomes will be 6 × 2 = 12. Listing them requires care: {1H, 1T, 2H, 2T, 3H, 3T, 4H, 4T, 5H, 5T, 6H, 6T}. Always read the experiment description carefully to avoid missing any part of the process.

Worked Examples from NCERT Ex 16.1

• Example 1: A coin is tossed three times. Describe the sample space. Step 1: Identify the actions. The experiment consists of three consecutive actions: tossing a coin, then a second time, then a third time. Each toss has two possible outcomes: Heads (H) or Tails (T). Step 2: Systematically list the outcomes. Let's use a tree-like thought process: - First toss is H: - Second toss can be H or T. - If second is H, third can be H or T. Outcomes: HHH, HHT. - If second is T, third can be H or T. Outcomes: HTH, HTT. - First toss is T: - Second toss can be H or T. - If second is H, third can be H or T. Outcomes: THH, THT. - If second is T, third can be H or T. Outcomes: TTH, TTT. Step 3: Combine all possibilities into a set. Collect all the unique outcomes from Step 2. Final Sample Space (S): {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
• Example 2: A die is rolled. If the outcome is an even number, a coin is tossed once. If the outcome is an odd number, the coin is tossed twice. Describe the sample space. Step 1: Analyze the conditional experiment. The experiment has two distinct paths based on the die roll. - Path 1: Die shows an even number {2, 4, 6}. Then, toss a coin once {H, T}. - Path 2: Die shows an odd number {1, 3, 5}. Then, toss a coin twice {HH, HT, TH, TT}. Step 2: List outcomes for Path 1. Combine each even number with the coin toss outcomes: - If die is 2, outcomes are 2H, 2T. - If die is 4, outcomes are 4H, 4T. - If die is 6, outcomes are 6H, 6T. Outcomes for this path: {2H, 2T, 4H, 4T, 6H, 6T} Step 3: List outcomes for Path 2. Combine each odd number with the two-coin-toss outcomes: - If die is 1, outcomes are 1HH, 1HT, 1TH, 1TT. - If die is 3, outcomes are 3HH, 3HT, 3TH, 3TT. - If die is 5, outcomes are 5HH, 5HT, 5TH, 5TT. Outcomes for this path: {1HH, 1HT, 1TH, 1TT, 3HH, 3HT, 3TH, 3TT, 5HH, 5HT, 5TH, 5TT} Step 4: Combine all outcomes from both paths. The total sample space is the union of the outcomes from Step 2 and Step 3. Final Sample Space (S): {2H, 2T, 4H, 4T, 6H, 6T, 1HH, 1HT, 1TH, 1TT, 3HH, 3HT, 3TH, 3TT, 5HH, 5HT, 5TH, 5TT}

Exam Traps and Common Mistakes

A very common mistake in Exercise 16.1 is confusing order. When two distinct dice are rolled, the outcome (1, 6) is different from (6, 1). However, if you are just asked to find the sum, the order might not matter depending on the question. For Ex 16.1, always assume order matters unless stated otherwise. For example, in a two-coin toss, HT (Heads first, Tails second) is a different outcome from TH (Tails first, Heads second). Listing just {HH, HT, TT} would be incorrect and would cost you marks. Another trap is incomplete listing. For complex experiments, students often miss one or two outcomes. Always double-check your total count using the multiplication principle if possible (e.g., 3 coin tosses = 2x2x2 = 8 outcomes) and make sure your listed set matches this count.

Practice Questions with Solutions

• Q: From a group of 2 boys (B1, B2) and 2 girls (G1, G2), a committee of two is to be formed. Describe the sample space. A: Step 1: Identify the items to be selected. We are selecting 2 people from a set of 4 {B1, B2, G1, G2}. Step 2: Systematically list all possible pairs, ensuring not to repeat a pair (since the order in a committee doesn't matter, {B1, G1} is the same as {G1, B1}). - Pair B1 with others: {B1, B2}, {B1, G1}, {B1, G2} - Pair B2 with others (not B1 as it's already listed): {B2, G1}, {B2, G2} - Pair G1 with others (not B1 or B2): {G1, G2} Step 3: Combine all unique pairs into a set. Final answer: S = {{B1, B2}, {B1, G1}, {B1, G2}, {B2, G1}, {B2, G2}, {G1, G2}}
• Q: An experiment consists of tossing a coin and then throwing a die only if a head comes up. Describe the sample space. A: Step 1: Analyze the experiment's structure. The experiment has two possible initial outcomes for the coin toss: Heads (H) or Tails (T). Step 2: Consider the case where the coin shows Tails (T). The experiment stops. So, 'T' is one of the final outcomes. Step 3: Consider the case where the coin shows Heads (H). The experiment continues by throwing a die. The possible outcomes for the die are {1, 2, 3, 4, 5, 6}. We must pair 'H' with each of these die outcomes. This gives the outcomes: H1, H2, H3, H4, H5, H6. Step 4: Combine the outcomes from all possible paths. Final answer: S = {T, H1, H2, H3, H4, H5, H6}
• Q: A coin is tossed. If it shows a head, we draw a ball from a bag consisting of 3 red (R1, R2, R3) and 2 black balls (B1, B2). If it shows a tail, we toss a die. Find the sample space. A: Step 1: Break down the experiment into two cases based on the initial coin toss. Case 1: Coin shows Head (H). We then draw a ball. The possible balls are {R1, R2, R3, B1, B2}. The outcomes for this case are {H R1, H R2, H R3, H B1, H B2}. Case 2: Coin shows Tail (T). We then toss a die. The possible outcomes for the die are {1, 2, 3, 4, 5, 6}. The outcomes for this case are {T1, T2, T3, T4, T5, T6}. Step 2: The total sample space is the collection of all outcomes from both cases. Final answer: S = {H R1, H R2, H R3, H B1, H B2, T1, T2, T3, T4, T5, T6}
• Q: Two dice (one red, one blue) are rolled. Write the sample space. A: Step 1: Understand that the dice are distinct (red and blue). This means the order of the numbers matters. For example, (Red=1, Blue=2) is a different outcome from (Red=2, Blue=1). Step 2: Let the outcome be represented by an ordered pair (x, y), where x is the number on the red die and y is the number on the blue die. Both x and y can take values from {1, 2, 3, 4, 5, 6}. Step 3: List the outcomes systematically. We can fix the outcome of the red die and list all possibilities for the blue die. - If Red=1, pairs are (1,1), (1,2), (1,3), (1,4), (1,5), (1,6) - If Red=2, pairs are (2,1), (2,2), (2,3), (2,4), (2,5), (2,6) - ... and so on, up to Red=6. - If Red=6, pairs are (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) Step 4: Describe the complete set. There will be 6 x 6 = 36 outcomes in total. Final answer: S = {(x, y) : x, y ∈ {1, 2, 3, 4, 5, 6}}. A full list would include all 36 pairs from (1,1) to (6,6).

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