Verify probability from independent trials

2 Alisha has four coins. One of these coins is biased so that the probability of obtaining a head is 0.6 . The other three coins are fair. Alisha throws the four coins at the same time. The random variable \(X\) denotes the number of heads obtained.

1. Show that the probability of obtaining exactly one head is 0.225 .
2. Complete the following probability distribution table for \(X\).

   \(x\)	0	1	2	3	4
   \(\mathrm { P } ( X = x )\)	0.05	0.225			0.075

3. Given that \(\mathrm { E } ( X ) = 2.1\), find the value of \(\operatorname { Var } ( X )\).
   \(380 \%\) of the residents of Kinwawa are in favour of a leisure centre being built in the town.
   20 residents of Kinwawa are chosen at random and asked, in turn, whether they are in favour of the leisure centre.
4. Find the probability that more than 17 of these residents are in favour of the leisure centre.
5. Find the probability that the 5th person asked is the first person who is not in favour of the leisure centre.
6. Find the probability that the 7th person asked is the second person who is not in favour of the leisure centre.

6 A fair tetrahedral die has four triangular faces, numbered \(1,2,3\) and 4 . The score when this die is thrown is the number on the face that the die lands on. This die is thrown three times. The random variable \(X\) is the sum of the three scores.

1. Show that \(\mathrm { P } ( X = 9 ) = \frac { 10 } { 64 }\).
2. Copy and complete the probability distribution table for \(X\).

   \(x\)	3	4	5	6	7	8	9	10	11	12
   \(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 64 }\)	\(\frac { 3 } { 64 }\)			\(\frac { 12 } { 64 }\)

3. Event \(R\) is 'the sum of the three scores is 9 '. Event \(S\) is 'the product of the three scores is 16 '. Determine whether events \(R\) and \(S\) are independent, showing your working.

1 Yasmin has 5 coins. One of these coins is biased with P (heads) \(= 0.6\). The other 4 coins are fair. She tosses all 5 coins once and records the number of heads, \(X\).

1. Show that \(\mathrm { P } ( X = 0 ) = 0.025\).
2. Show that \(\mathrm { P } ( X = 1 ) = 0.1375\). The table shows the probability distribution of \(X\).

   \(r\)	0	1
   \(\mathrm { P } ( X = r )\)	0.025	0.1375	0.3	0.325	0.175	0.0375

3. Draw a vertical line chart to illustrate the probability distribution.
4. Comment on the skewness of the distribution.
5. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
6. Yasmin tosses the 5 coins three times. Find the probability that the total number of heads is 3 .

2 Three fair six-sided dice are thrown. The random variable \(X\) represents the highest of the three scores on the dice.

1. Show that \(\mathrm { P } ( X = 6 ) = \frac { 91 } { 216 }\). The table shows the probability distribution of \(X\).

   \(r\)	1	2	3	4	5	6
   \(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 216 }\)	\(\frac { 7 } { 216 }\)	\(\frac { 19 } { 216 }\)	\(\frac { 37 } { 216 }\)	\(\frac { 61 } { 216 }\)	\(\frac { 91 } { 216 }\)

2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

5 Yasmin has 5 coins. One of these coins is biased with P (heads) \(= 0.6\). The other 4 coins are fair. She tosses all 5 coins once and records the number of heads, \(X\).

1. Show that \(\mathrm { P } ( X = 0 ) = 0.025\).
2. Show that \(\mathrm { P } ( X = 1 ) = 0.1375\). The table shows the probability distribution of \(X\).

   \(r\)	0	1	2	3	4	5
   \(\mathrm { P } ( X = r )\)	0.025	0.1375	0.3	0.325	0.175	0.0375

3. Draw a vertical line chart to illustrate the probability distribution.
4. Comment on the skewness of the distribution.
5. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
6. Yasmin tosses the 5 coins three times. Find the probability that the total number of heads is 3 .

1 Each time a certain triangular spinner is spun, it lands on one of the numbers 0,1 and 2 with probabilities as shown in the table. The spinner is spun twice. The total of the two numbers on which it lands is denoted by \(X\).

1. Show that \(\mathrm { P } ( X = 2 ) = 0.18\). The probability distribution of \(X\) is given in the table.

   \(x\)	0	1	2	3	4
   \(\mathrm { P } ( X = x )\)	0.49	0.28	0.18	0.04	0.01

2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

7 Yasmin has 5 coins. One of these coins is biased with P (heads) \(= 0.6\). The other 4 coins are fair. She tosses all 5 coins once and records the number of heads, \(X\).

1. Show that \(\mathrm { P } ( X = 0 ) = 0.025\).
2. Show that \(\mathrm { P } ( X = 1 ) = 0.1375\). The table shows the probability distribution of \(X\).

   \(r\)	0	1	2	3	4	5
   \(\mathrm { P } ( X = r )\)	0.025	0.1375	0.3	0.325	0.175	0.0375

3. Draw a vertical line chart to illustrate the probability distribution.
4. Comment on the skewness of the distribution.
5. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
6. Yasmin tosses the 5 coins three times. Find the probability that the total number of heads is 3 . \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

11 Alex models the number of goals that a local team will score in any match as follows. The number of goals scored in any match is independent of the number of goals scored in any other match.

1. Alex chooses 3 matches at random. Use the model to determine the probability of each of the following.
   1. The team will score a total of exactly 1 goal in the 3 matches.
   2. The numbers of goals scored in the first 2 of the 3 matches will be equal, but the number of goals scored in the 3rd match will be different. During the first 10 matches this season, the team scores a total of 31 goals.
2. Without carrying out a formal test, explain briefly whether this casts doubt on the validity of Alex's model. \section*{END OF QUESTION PAPER}

7. Tetrahedral dice have four faces. Two fair tetrahedral dice, one red and one blue, have faces numbered \(0,1,2\), and 3 respectively. The dice are rolled and the numbers face down on the two dice are recorded. The random variable \(R\) is the score on the red die and the random variable \(B\) is the score on the blue die.

1. Find \(\mathrm { P } ( R = 3\) and \(B = 0 )\). The random variable \(T\) is \(R\) multiplied by \(B\).
2. Complete the diagram below to represent the sample space that shows all the possible values of \(T\).
   \includegraphics[max width=\textwidth, alt={}, center]{af84d17b-5308-4b1e-99b5-40c5df5bf01e-13_732_771_834_621} \section*{Sample space diagram of \(T\)}
3. The table below represents the probability distribution of the random variable \(T\).

   \(t\)	0	1	2	3	4	6	9
   \(\mathrm { P } ( T = t )\)	\(a\)	\(b\)	\(1 / 8\)	\(1 / 8\)	\(c\)	\(1 / 8\)	\(d\)

   Find the values of \(a , b , c\) and \(d\). Find the values of
4. \(\mathrm { E } ( T )\),
5. \(\operatorname { Var } ( T )\).

1 In a quiz a contestant is asked up to four questions. The contestant's turn ends once the contestant gets a question wrong or has answered all four questions. The probability that a particular contestant gets any question correct is 0.6 , independently of other questions. The discrete random variable \(X\) models the number of questions which the contestant gets correct in a turn.

1. Show that \(\mathrm { P } ( X = 4 ) = 0.1296\). The probability distribution of \(X\) is shown in Fig. 1.1. \begin{table}[h]

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	0.4	0.24	0.144	0.0864	0.1296

   \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{table}
2. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   The number of points that a contestant scores is as shown in Fig. 1.2. \begin{table}[h]

   Number of questions correct	Number of points scored
   0 or 1	0
   2	2
   3	3
   4	5

   \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{table} The discrete random variable \(Y\) models the number of points which the contestant scores.
3. Without doing any working, explain whether each of the following will be less than, equal to or greater than the corresponding value for \(X\).
   • \(\mathrm { E } ( Y )\)
   • \(\operatorname { Var } ( Y )\)

17 A game consists of spinning a circular wheel divided into numbered sectors as shown below.
\includegraphics[max width=\textwidth, alt={}, center]{7cca79eb-fd09-4ec0-8a1d-a7a38ca73f7a-22_764_963_404_541} On each spin the score, \(X\), is the value shown in the sector that the arrow points to when the spinner stops. The probability of the arrow pointing at a sector is proportional to the angle subtended at the centre by that sector. 17

1. Show that \(\mathrm { P } ( X = 1 ) = \frac { 5 } { 18 }\)
   17
2. Complete the probability distribution for \(X\) in the table below.

   \(\boldsymbol { x }\)	1
   \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	\(\frac { 5 } { 18 }\)