Multi-stage game or match outcomes

6 Sajid is practising for a long jump competition. He counts any jump that is longer than 6 m as a success. On any day, the probability that he has a success with his first jump is 0.2 . For any subsequent jump, the probability of a success is 0.3 if the previous jump was a success and 0.1 otherwise. Sajid makes three jumps.

1. Draw a tree diagram to illustrate this information, showing all the probabilities.
2. Find the probability that Sajid has exactly one success given that he has at least one success.
   On another day, Sajid makes six jumps.
3. Find the probability that only his first three jumps are successes or only his last three jumps are successes.

3 Roger and Andy play a tennis match in which the first person to win two sets wins the match. The probability that Roger wins the first set is 0.6 . For sets after the first, the probability that Roger wins the set is 0.7 if he won the previous set, and is 0.25 if he lost the previous set. No set is drawn.

1. Find the probability that there is a winner of the match after exactly two sets.
2. Find the probability that Andy wins the match given that there is a winner of the match after exactly two sets.

3 Redbury United soccer team play a match every week. Each match can be won, drawn or lost. At the beginning of the soccer season the probability that Redbury United win their first match is \(\frac { 3 } { 5 }\), with equal probabilities of losing or drawing. If they win the first match, the probability that they win the second match is \(\frac { 7 } { 10 }\) and the probability that they lose the second match is \(\frac { 1 } { 10 }\). If they draw the first match they are equally likely to win, draw or lose the second match. If they lose the first match, the probability that they win the second match is \(\frac { 3 } { 10 }\) and the probability that they draw the second match is \(\frac { 1 } { 20 }\).

1. Draw a fully labelled tree diagram to represent the first two matches played by Redbury United in the soccer season.
2. Given that Redbury United win the second match, find the probability that they lose the first match.

5 Rachel and Anna play each other at badminton. Each game results in either a win for Rachel or a win for Anna. The probability of Rachel winning the first game is 0.6 . If Rachel wins a particular game, the probability of her winning the next game is 0.7 , but if she loses, the probability of her winning the next game is 0.4 . By using a tree diagram, or otherwise,

1. find the conditional probability that Rachel wins the first game, given that she loses the second,
2. find the probability that Rachel wins 2 games and loses 1 game out of the first three games they play.