Probability distribution table

6 Eli has four fair 4 -sided dice with sides labelled \(1,2,3,4\). He throws all four dice at the same time. The random variable \(X\) denotes the number of 2s obtained.

1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 3 } { 64 }\).
2. Complete the following probability distribution table for \(X\).

   \(x\)	0	1	2	3	4
   \(\mathrm { P } ( X = x )\)	\(\frac { 81 } { 256 }\)			\(\frac { 3 } { 64 }\)	\(\frac { 1 } { 256 }\)

3. Find \(\mathrm { E } ( X )\).
   Eli throws the four dice at the same time on 96 occasions.
4. Use an approximation to find the probability that he obtains at least two 2 s on fewer than 20 of these occasions.

3 A fair dice has four faces. One face is coloured pink, one is coloured orange, one is coloured green and one is coloured black. Five such dice are thrown and the number that fall on a green face are counted. The random variable \(X\) is the number of dice that fall on a green face.

1. Show that the probability of 4 dice landing on a green face is 0.0146 , correct to 4 decimal places.
2. Draw up a table for the probability distribution of \(X\), giving your answers correct to 4 decimal places.

3. In a town, \(30 \%\) of residents listen to the local radio station. Four residents are chosen at random.

1. State the distribution of the random variable \(X\), the number of these four residents that listen to local radio.
2. On graph paper, draw the probability distribution of \(X\).
3. Write down the most likely number of these four residents that listen to the local radio station.
4. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

7 On a multiple choice examination paper, each question has five alternative answers given, only one of which is correct. For each question, candidates gain 4 marks for a correct answer but lose 1 mark for an incorrect answer.

1. James guesses the answer to each question.
   1. Copy and complete the following table for the probability distribution of \(X\), the number of marks obtained by James for each question.

      \(\boldsymbol { x }\)	4	- 1
      \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)

   2. Hence find \(\mathrm { E } ( X )\).
2. Karen is able to eliminate two of the incorrect answers from the five alternative answers given for each question before guessing the answer from those remaining. Given that the examination paper contains 24 questions, calculate Karen's expected total mark.