Calculate simple probabilities

3. A machine pours oil into bottles. It is electronically controlled to cut off the flow of oil randomly between 100 ml and \(k \mathrm { ml }\), where \(k > 100\). It is equally likely to cut off the flow at any point in this range. The random variable \(X\) is the volume of oil poured into a bottle. Given that \(\mathrm { P } ( 102 \leqslant X \leqslant k ) = \frac { 2 } { 3 }\)

1. show that \(k = 106\)
2. Find the probability that the volume of oil poured into a bottle is
   1. less than 105 ml ,
   2. exactly 105 ml .
3. Write down the value of \(\mathrm { E } ( X )\)
4. Find the 15th percentile of this distribution.
5. Determine the value of \(x\) such that \(3 \mathrm { P } ( X \leqslant x - 1.5 ) = \mathrm { P } ( X \geqslant x + 1.5 )\)

1. A point is to be randomly plotted on the \(x\)-axis, where the units are measured in cm .

The random variable \(R\) represents the \(x\) coordinate of the point on the \(x\)-axis and \(R\) is uniformly distributed over the interval [-5,19] A negative value indicates that the point is to the left of the origin and a positive value indicates that the point is to the right of the origin.

1. Find the exact probability that the point is plotted to the right of the origin.
2. Find the exact probability that the point is plotted more than 3.5 cm away from the origin.
3. Sketch the cumulative distribution function of \(R\) Three independent points with \(x\) coordinates \(R _ { 1 } , R _ { 2 }\) and \(R _ { 3 }\) are plotted on the \(x\)-axis.
4. Find the exact probability that
   1. all three points are more than 10 cm from the origin
   2. the point furthest from the origin is more than 10 cm from the origin.

3. In an old computer game a white square representing a ball appears at random at the top of the playing area, which is 24 cm wide, and moves down the screen. The continuous random variable \(X\) represents the distance, in centimetres, of the dot from the left-hand edge of the screen when it appears. The distribution of \(X\) is rectangular over the interval [4,28].

1. Find the mean and variance of \(X\).
2. Find \(\mathrm { P } ( | X - 16 | < 3 )\). During a single game, a player receives 12 "balls".
3. Find the probability that the ball appears within 3 cm of the middle of the top edge of the playing area more than four times in a single game.
   (3 marks)

1 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 5 } & 1 \leq x \leq 6 \\ 0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { P } ( X \geq 3 )\) Circle your answer. \(\frac { 1 } { 5 } \quad \frac { 2 } { 5 } \quad \frac { 3 } { 5 } \quad \frac { 4 } { 5 }\)

9 Lianne models the maximum time in hours that a rechargeable battery can be used, before needing to be recharged, with a rectangular distribution with values between 8 and 12 9

1. The probability that the maximum time the battery can be used before needing to be recharged is more than 10.5 hours is equal to \(p\) Lianne will only buy the battery if \(p\) is more than 0.4
   Determine whether Lianne will buy the battery.
   [0pt] [2 marks]
   9
2. A histogram is plotted for 100 recharges showing the maximum time the battery can be used before needing to be recharged. \includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-15_670_1186_404_427} Explain why the model used in part (a) may not be valid and suggest the name of a different distribution that could be used to model the maximum time between recharges. \includegraphics[max width=\textwidth, alt={}, center]{62cee897-6eac-40b3-84c1-a0d165ba6903-16_2488_1732_219_139}
   \includegraphics[max width=\textwidth, alt={}]{62cee897-6eac-40b3-84c1-a0d165ba6903-20_2496_1721_214_148}