Question: In the semi-finals of a padel tournament, Andrew will play Beth and Connie will play Daniel. The two winners will then play each other in the final. The probability that Andrew will win against Beth is $0.3$. The probability that Andrew will win against Connie is $0.8$. The probability that Andrew will win against Daniel is $0.4$. The probability that Connie will win against Daniel is $0.5$. Work out the probability that Andrew will win the padel tournament.
1. **State the problem:** We want to find the probability that Andrew wins the entire tournament.
2. **Identify matches and probabilities:**
- Semi-final 1: Andrew vs Beth, $P(A\text{ beats }B) = 0.3$
- Semi-final 2: Connie vs Daniel, $P(C\text{ beats }D) = 0.5$
- Final: Andrew plays the winner of Connie vs Daniel.
3. **Possible final opponents for Andrew:**
- If Connie wins, Andrew plays Connie with $P(A\text{ beats }C) = 0.8$
- If Daniel wins, Andrew plays Daniel with $P(A\text{ beats }D) = 0.4$
4. **Calculate probability Andrew reaches final:**
$$P(\text{Andrew in final}) = P(A\text{ beats }B) = 0.3$$
5. **Calculate probability Andrew wins final:**
- Andrew wins final if he beats Connie and Connie wins semi-final:
$$P(A\text{ beats }C \text{ and } C\text{ beats }D) = P(A\text{ beats }C) \times P(C\text{ beats }D) = 0.8 \times 0.5 = 0.4$$
- Andrew wins final if he beats Daniel and Daniel wins semi-final:
$$P(A\text{ beats }D \text{ and } D\text{ beats } C) = P(A\text{ beats }D) \times (1 - P(C\text{ beats }D)) = 0.4 \times (1 - 0.5) = 0.4 \times 0.5 = 0.2$$
6. **Total probability Andrew wins final given he is in final:**
$$P(\text{Andrew wins final}) = 0.4 + 0.2 = 0.6$$
7. **Calculate total probability Andrew wins tournament:**
$$P(\text{Andrew wins tournament}) = P(\text{Andrew in final}) \times P(\text{Andrew wins final}) = 0.3 \times 0.6 = 0.18$$
**Final answer:**
$$\boxed{0.18}$$
Andrew has an 18% chance of winning the tournament.