Draw a tree diagram for three fair coins tossed together once. a) Derive the sample space. b) Find the probability of getting: i) at least one head ii) at most one tail iii) exactly three heads.

Here's the breakdown for tossing three fair coins: a) Draw a tree diagram for three fair coins tossed together once. Derive the sample space. Tree Diagram Description: Start with a single point. 1. From the starting point, draw two branches for the first coin: one labeled 'H' (Heads) and one labeled 'T' (Tails). 2. From the end of each of these branches, draw two more branches for the second coin: 'H' and 'T'. This gives paths like HH, HT, TH, TT. 3. From the end of each of these four branches, draw two final branches for the third coin: 'H' and 'T'. This will lead to eight final outcomes. Sample Space: The sample space (S) lists all possible outcomes when three fair coins are tossed. S = \HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\ The total number of possible outcomes is n(S) = 8. b) Find the probability of getting: i) at least one head Step 1: Identify outcomes with at least one head. This means outcomes with one, two, or three heads. The outcomes are: HHH, HHT, HTH, HTT, THH, THT, TTH. The number of favorable outcomes is 7. Step 2: Calculate the probability. P(at least one head) = Number of outcomes with at least one headTotal number of outcomes = (7)/(8) The probability of getting at least one head is (7)/(8). ii) at most one tail Step 1: Identify outcomes with at most one tail. This means outcomes with zero tails or one tail. • Outcomes with zero tails: HHH (1 outcome) • Outcomes with one tail: HHT, HTH, THH (3 outcomes) The total number of favorable outcomes is 1 + 3 = 4. Step 2: Calculate the probability. P(at most one tail) = Number of outcomes with at most one tailTotal number of outcomes = (4)/(8) = (1)/(2) The probability of getting at most one tail is (1)/(2). iii) exactly three heads Step 1: Identify outcomes with exactly three heads. The only outcome with exactly three heads is HHH. The number of favorable outcomes is 1. Step 2: Calculate the probability. P(exactly three heads) = Number of outcomes with exactly three headsTotal number of outcomes = (1)/(8) The probability of getting exactly three heads is (1)/(8). 3 done, 2 left today. You're making progress.