Edexcel S1 (Statistics 1) 2023 October

1. Sally plays a game in which she can either win or lose.

A turn consists of up to 3 games. On each turn Sally plays the game up to 3 times. If she wins the first 2 games or loses the first 2 games, then she will not play the 3rd game.

• The probability that Sally wins the first game in a turn is 0.7
• If Sally wins a game the probability that she wins the next game is 0.6
• If Sally loses a game the probability that she wins the next game is 0.2
  1. Use this information to complete the tree diagram on page 3
  2. Find the probability that Sally wins the first 2 games in a turn.
  3. Find the probability that Sally wins exactly 2 games in a turn.

Given that Sally wins 2 games in a turn,

find the probability that she won the first 2 games. Given that Sally won the first game in a turn,

find the probability that she won 2 games. 1st game 2nd game win

1. The weights, to the nearest kilogram, of a sample of 33 red kangaroos taken in December are summarised in the stem and leaf diagram below.

Key: 3 | 2 represents 32 kg

1. Find
   1. the value of the median
   2. the value of \(Q _ { 1 }\) and the value of \(Q _ { 3 }\)
      for the weights of these red kangaroos. For these data an outlier is defined as a value that is
      greater than \(Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\)
      or smaller than \(Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\)
2. Show that there are 2 outliers for these data. Figure 1 on page 7 shows a box plot for the weights of the same 33 red kangaroos taken in February, earlier in the year.
3. In the space on Figure 1, draw a box plot to represent the weights of these red kangaroos in December.
4. Compare the distribution of the weights of red kangaroos taken in February with the distribution of the weights of red kangaroos taken in December of the same year. You should interpret your comparisons in the context of the question.
   \includegraphics[max width=\textwidth, alt={}]{f94b29e0-081f-45e8-99a7-ac835eec91e5-07_2267_51_307_36}
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f94b29e0-081f-45e8-99a7-ac835eec91e5-07_766_1803_1777_132} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Turn over for a spare grid if you need to redraw your box plot. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f94b29e0-081f-45e8-99a7-ac835eec91e5-09_901_1833_1653_114} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} \begin{verbatim} (Total for Question 2 is 13 marks) \end{verbatim}

1. 1. Bob shops at a market each week. The event that

Bob buys carrots is denoted by \(C\)
Bob buys onions is denoted by \(O\)
At each visit, Bob may buy neither, or one, or both of these items. The probability that Bob buys carrots is 0.65
Bob does not buy onions is 0.3
Bob buys onions but not carrots is 0.15
The Venn diagram below represents the events \(C\) and \(O\)
\includegraphics[max width=\textwidth, alt={}, center]{f94b29e0-081f-45e8-99a7-ac835eec91e5-10_453_851_877_607}
where \(w , x , y\) and \(z\) are probabilities.

1. Find the value of \(w\), the value of \(x\), the value of \(y\) and the value of \(z\) For one visit to the market,
2. find the probability that Bob buys either carrots or onions but not both.
3. Show that the events \(C\) and \(O\) are not independent.
   (ii) \(F , G\) and \(H\) are 3 events. \(F\) and \(H\) are mutually exclusive. \(F\) and \(G\) are independent. Given that $$\mathrm { P } ( F ) = \frac { 2 } { 7 } \quad \mathrm { P } ( H ) = \frac { 1 } { 4 } \quad \mathrm { P } ( F \cup G ) = \frac { 5 } { 8 }$$
4. find \(\operatorname { P } ( F \cup H )\)
5. find \(\mathrm { P } ( G )\)
6. find \(\operatorname { P } ( F \cap G )\)

1. The discrete random variable \(X\) has the following probability distribution.

1. Show that \(\mathrm { E } \left( \frac { 1 } { X } \right) = \frac { 2 } { 5 }\)
2. Find \(\operatorname { Var } \left( \frac { 1 } { X } \right)\) The random variable \(Y = \frac { 30 } { X }\)
3. Find
   1. \(\mathrm { E } ( Y )\)
   2. \(\operatorname { Var } ( Y )\)
4. Find \(\mathrm { P } ( X < 3 \mid Y < 20 )\)

1. The weights, \(X\) grams, of a particular variety of fruit are normally distributed with

$$X \sim \mathrm {~N} \left( 210,25 ^ { 2 } \right)$$ A fruit of this variety is selected at random.

1. Show that the probability that the weight of this fruit is less than 240 grams is 0.8849
2. Find the probability that the weight of this fruit is between 190 grams and 240 grams.
3. Find the value of \(k\) such that \(\mathrm { P } ( 210 - k < X < 210 + k ) = 0.95\) A wholesaler buys large numbers of this variety of fruit and classifies the lightest \(15 \%\) as small.
4. Find the maximum weight of a fruit that is classified as small. You must show your working clearly. The weights, \(Y\) grams, of a second variety of this fruit are normally distributed with $$Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$$ Given that \(5 \%\) of these fruit weigh less than 152 grams and \(40 \%\) weigh more than 180 grams,
5. calculate the mean and standard deviation of the weights of this variety of fruit.

1. The variables \(x\) and \(y\) have the following regression equations based on the same 12 observations.

1. 1. Find the point of intersection of these lines.
   2. Hence show that \(\sum x = 25\) Given that $$\sum x y = \frac { 6961 } { 60 }$$
2. Find \(S _ { x y }\)
3. Find the product moment correlation coefficient between \(x\) and \(y\)