Binomial with complementary events

4 A game for two players is played using a fair 4-sided dice with sides numbered 1, 2, 3 and 4. One turn consists of throwing the dice repeatedly up to a maximum of three times. When a 4 is obtained, no further throws are made during that turn. A player who obtains a 4 in their turn scores 1 point.

1. Show that the probability that a player obtains a 4 in one turn is \(\frac { 37 } { 64 }\).
   Xeno and Yao play this game.
2. Find the probability that neither Xeno nor Yao score any points in their first two turns.
3. Xeno and Yao each have three turns. Find the probability that Xeno scores 2 more points than Yao. \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_70_1548_376_349} \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_67_1566_466_328} ........................................................................................................................................ ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_72_1570_735_324} \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_72_1570_826_324} \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_77_1570_913_324} ........................................................................................................................................ . ......................................................................................................................................... \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_70_1570_1187_324} \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_67_1570_1279_324} ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_62_1570_1462_324} .......................................................................................................................................... ......................................................................................................................................... . .......................................................................................................................................... .......................................................................................................................................... . \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_71_1570_1905_324} \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_74_1570_1994_324} \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_76_1570_2083_324} ........................................................................................................................................ ........................................................................................................................................ \includegraphics[max width=\textwidth, alt={}]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_74_1570_2359_324} ......................................................................................................................................... \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_70_1570_2542_324} \includegraphics[max width=\textwidth, alt={}, center]{a909cef1-8a22-4cef-b0b7-c48316304c0c-07_70_1570_2631_324}

Mark is playing solitaire on his computer. The probability that he wins a game is 0.2, independently of all other games that he plays.

1. Find the expected number of wins in 12 games. [2]
2. Find the probability that
   1. he wins exactly 2 out of the next 12 games that he plays, [3]
   2. he wins at least 2 out of the next 12 games that he plays. [3]
3. Mark's friend Ali also plays solitaire. Ali claims that he is better at winning games than Mark. In a random sample of 20 games played by Ali, he wins 7 of them. Write down suitable hypotheses for a test at the 5\% level to investigate whether Ali is correct. Give a reason for your choice of alternative hypothesis. Carry out the test. [9]

Six standard dice with faces numbered 1 to 6 are thrown together. Assuming that the dice are fair, find the probability that

1. none of the dice show a score of 6, [3 marks]
2. more than one of the dice shows a score of 6, [4 marks]
3. there are equal numbers of odd and even scores showing on the dice. [3 marks]

One of the dice is suspected of being biased such that it shows a score of 6 more often than the other numbers. This die is thrown eight times and gives a score of 6 three times.

1. Stating your hypotheses clearly, test at the 5% level of significance whether or not this die is biased towards scoring a 6. [7 marks]