Probability distribution from tree

6 Every day Eduardo tries to phone his friend. Every time he phones there is a \(50 \%\) chance that his friend will answer. If his friend answers, Eduardo does not phone again on that day. If his friend does not answer, Eduardo tries again in a few minutes' time. If his friend has not answered after 4 attempts, Eduardo does not try again on that day.

1. Draw a tree diagram to illustrate this situation.
2. Let \(X\) be the number of unanswered phone calls made by Eduardo on a day. Copy and complete the table showing the probability distribution of \(X\).

   \(x\)	0	1	2	3	4
   \(\mathrm { P } ( X = x )\)		\(\frac { 1 } { 4 }\)

3. Calculate the expected number of unanswered phone calls on a day.

5 Maryam has 7 sweets in a tin; 6 are toffees and 1 is a chocolate. She chooses one sweet at random and takes it out. Her friend adds 3 chocolates to the tin. Then Maryam takes another sweet at random out of the tin.

1. Draw a fully labelled tree diagram to illustrate this situation.
2. Draw up the probability distribution table for the number of toffees taken.
3. Find the mean number of toffees taken.
4. Find the probability that the first sweet taken is a chocolate, given that the second sweet taken is a toffee.

6 Nadia is very forgetful. Every time she logs in to her online bank she only has a \(40 \%\) chance of remembering her password correctly. She is allowed 3 unsuccessful attempts on any one day and then the bank will not let her try again until the next day.

1. Draw a fully labelled tree diagram to illustrate this situation.
2. Let \(X\) be the number of unsuccessful attempts Nadia makes on any day that she tries to log in to her bank. Copy and complete the following table to show the probability distribution of \(X\).

   \(x\)	0	1	2	3
   \(\mathrm { P } ( X = x )\)		0.24

3. Calculate the expected number of unsuccessful attempts made by Nadia on any day that she tries to \(\log\) in.

6 Nadia is very forgetful. Every time she logs in to her online bank she only has a \(40 \%\) chance of remembering her password correctly. She is allowed 3 unsuccessful attempts on any one day and then the bank will not let her try again until the next day.

1. Draw a fully labelled tree diagram to illustrate this situation.
2. Let \(X\) be the number of unsuccessful attempts Nadia makes on any day that she tries to log in to her bank. Complete the following table to show the probability distribution of \(X\).

   \(x\)	0	1	2	3
   \(\mathrm { P } ( X = x )\)		0.24

3. Calculate the expected number of unsuccessful attempts made by Nadia on any day that she tries to \(\log\) in.

1. A bag contains a large number of counters with one of the numbers 5 , 10 or 20 written on each of them in the ratio \(5 : 2 : a\)

A jar contains a large number of counters with one of the numbers 5 or 10 written on each of them in the ratio \(1 : 3\) One counter is selected at random from the bag and then two counters are selected at random from the jar.
The random variable \(R\) represents the range of the numbers on the 3 counters.
Given that \(\mathrm { P } ( R = 15 ) = \frac { 63 } { 256 }\)

1. by forming and solving an equation in \(a\), show that \(a = 9\)
2. find the sampling distribution of \(R\)