Two unbiased coins are tossed, and we need to complete the outcome table and calculate the probability of three events: (i) getting two heads, (ii) getting different outcomes, and (iii) getting at least one tail.

1. Problem Description

Two unbiased coins are tossed, and we need to complete the outcome table and calculate the probability of three events: (i) getting two heads, (ii) getting different outcomes, and (iii) getting at least one tail.

2. Solution Steps

(a) Complete the outcome table:

The table rows represent the outcomes of Coin 2 (H or T), and the columns represent the outcomes of Coin 1 (H or T). The entries in the table represent the combination of outcomes for Coin 1 and Coin

2. - When Coin 2 is H and Coin 1 is T, the outcome is HT.

- When Coin 2 is T and Coin 1 is H, the outcome is TH.

The completed table is:

| | Coin 1 | |

|--------|--------|-------|

| | H | T |

| Coin 2 | H | HH | HT |

| | T | TH | TT |

(b) Calculate probabilities:

(i) Probability of getting 2 heads:

From the table, there is only one outcome with two heads (HH). The total number of outcomes is 4 (HH, HT, TH, TT).

Therefore, the probability of getting two heads is:

$P(\text{2 heads}) = \frac{\text{Number of outcomes with 2 heads}}{\text{Total number of outcomes}} = \frac{1}{4}$

(ii) Probability of getting different outcomes:

Different outcomes mean one head and one tail (HT or TH).

From the table, there are two outcomes with different results (HT, TH).

Therefore, the probability of getting different outcomes is:

$P(\text{different outcomes}) = \frac{\text{Number of outcomes with different results}}{\text{Total number of outcomes}} = \frac{2}{4} = \frac{1}{2}$

(iii) Probability of getting at least one tail:

"At least one tail" means one tail or two tails.

The outcomes with at least one tail are HT, TH, and TT.

Therefore, the probability of getting at least one tail is:

$P(\text{at least one tail}) = \frac{\text{Number of outcomes with at least one tail}}{\text{Total number of outcomes}} = \frac{3}{4}$

3. Final Answer

(a) The completed outcome table is:

| | Coin 1 | |

|--------|--------|-------|

| | H | T |

| Coin 2 | H | HH | HT |

| | T | TH | TT |

(b) The probabilities are:

(i) Probability of getting 2 heads: $\frac{1}{4}$

(ii) Probability of getting different outcomes: $\frac{1}{2}$

(iii) Probability of getting at least one tail: $\frac{3}{4}$