Geometric then binomial separate scenarios

3 Kayla is competing in a throwing event. A throw is counted as a success if the distance achieved is greater than 30 metres. The probability that Kayla will achieve a success on any throw is 0.25 .

1. Find the probability that Kayla takes more than 6 throws to achieve a success.
2. Find the probability that, for a random sample of 10 throws, Kayla achieves at least 3 successes.

1 A fair six-sided die, with faces marked \(1,2,3,4,5,6\), is thrown repeatedly until a 4 is obtained.

1. Find the probability that obtaining a 4 requires fewer than 6 throws.
   On another occasion, the die is thrown 10 times.
2. Find the probability that a 4 is obtained at least 3 times.

2 Hazeem repeatedly throws two ordinary fair 6-sided dice at the same time. On each occasion, the score is the sum of the two numbers that she obtains.

1. Find the probability that it takes exactly 5 throws of the two dice for Hazeem to obtain a score of 8 or more.
2. Find the probability that it takes no more than 4 throws of the two dice for Hazeem to obtain a score of 8 or more.
3. For 8 randomly chosen throws of the two dice, find the probability that Hazeem obtains a score of 8 or more on fewer than 3 occasions.

2 George has a fair 5 -sided spinner with sides labelled 1,2,3,4,5. He spins the spinner and notes the number on the side on which the spinner lands.

1. Find the probability that it takes fewer than 7 spins for George to obtain a 5 .
   George spins the spinner 10 times.
2. Find the probability that he obtains a 5 more than 4 times but fewer than 8 times.

5 The probability that a driver passes an advanced driving test is 0.3 on any given attempt.

1. Dipak keeps taking the test until he passes. The random variable \(X\) denotes the number of attempts required for Dipak to pass the test.
   1. Find \(\mathrm { P } ( 2 \leqslant X \leqslant 6 )\).
   2. Find \(\mathrm { E } ( X )\).
      Five friends will each take their advanced driving test tomorrow.
2. Find the probability that at least three of them will pass tomorrow.
   75 people will take their advanced driving test next week.
   [0pt]
3. Use an approximation to find the probability that more than 20 of them will pass next week. [5]

4. (a) Write down the conditions under which the binomial distribution may be a suitable model to use in statistical work. A six-sided die is biased. When the die is thrown the number 5 is twice as likely to appear as any other number. All the other faces are equally likely to appear. The die is thrown repeatedly. Find the probability that
(b) (i) the first 5 will occur on the sixth throw,
(ii) in the first eight throws there will be exactly three 5 s .