Probability Definitions

The probabilities of events \(A\), \(B\) and \(C\) are related, as shown in the Venn diagram below. \includegraphics{figure_10} Find the value of \(x\). Circle your answer. [1 mark] \(0.11\) \quad \(0.46\) \quad \(0.54\) \quad \(0.89\)

The probabilities of events \(A\), \(B\) and \(C\) are related, as shown in the Venn diagram below. \includegraphics{figure_10} Find the value of \(x\). Circle your answer. [1 mark] \(0.11\) \quad \(0.46\) \quad \(0.54\) \quad \(0.89\)

9. The Venn diagram, where \(p , q\) and \(r\) are probabilities, shows the events \(A , B , C\) and \(D\) and associated probabilities. \includegraphics[max width=\textwidth, alt={}, center]{fdff6575-679e-4d25-ad43-e9d343c1746f-22_623_1130_326_438}

1. State any pair of mutually exclusive events from \(A\), \(B\), \(C\) and \(D\) The events \(B\) and \(C\) are independent.
2. Find the value of \(p\)
3. Find the greatest possible value of \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\) Given that \(\mathrm { P } \left( B \mid A ^ { \prime } \right) = 0.5\)
4. find the value of \(q\) and the value of \(r\)
5. Find \(\mathrm { P } \left( [ A \cup B ] ^ { \prime } \cap C \right)\)
6. Use set notation to write an expression for the event with probability \(p\)

The events \(A, B\) are such that \(P(A) = 0.2, P(B) = 0.3\). Determine the value of \(P(A \cup B)\) when

1. \(A,B\) are mutually exclusive, [2]
2. \(A,B\) are independent, [3]
3. \(A \subset B\). [1]

An architect bids for two construction projects. He estimates the probability of winning bid \(A\) is \(0 \cdot 6\), the probability of winning bid \(B\) is \(0 \cdot 5\) and the probability of winning both is \(0 \cdot 2\).

1. Show that the probability that he does not win either bid is \(0 \cdot 1\). [2]
2. Find the probability that he wins exactly one bid. [2]
3. Given that he does not win bid \(A\), find the probability that he wins bid \(B\). [3]

1. The events \(A\) and \(B\) satisfy

$$\mathrm { P } ( A ) = x \quad \mathrm { P } ( B ) = y \quad \mathrm { P } ( A \cup B ) = 0.65 \quad \mathrm { P } ( B \mid A ) = 0.3$$

1. Show that $$14 x + 20 y = 13$$ The events \(B\) and \(C\) are mutually exclusive such that $$\mathrm { P } ( B \cup C ) = 0.85 \quad \mathrm { P } ( C ) = \frac { 1 } { 2 } x + y$$
2. 1. Find a second equation in \(x\) and \(y\)
   2. Hence find the value of \(x\) and the value of \(y\)
3. Determine whether or not \(A\) and \(B\) are statistically independent. You must show your working clearly.

In lines 59 and 60, the text says "In that case the proportion suffering such an attack would be 6.4%." Explain how this figure was obtained. [1]

4 For a group of 250 cars the numbers, classified by colour and country of manufacture, are shown in the table. One car is selected at random from this group. Find the probability that the selected car is

1. a red or silver car manufactured in Korea,
2. not manufactured in Japan. \(X\) is the event that the selected car is white. \(Y\) is the event that the selected car is manufactured in Germany.
3. By using appropriate probabilities, determine whether events \(X\) and \(Y\) are independent.

5 Dafydd, Eli and Fabio are members of an amateur cycling club that holds a time trial each Sunday during the summer. The independent probabilities that Dafydd, Eli and Fabio take part in any one of these trials are \(0.6,0.7\) and 0.8 respectively. Find the probability that, on a particular Sunday during the summer:

1. none of the three cyclists takes part;
2. Fabio is the only one of the three cyclists to take part;
3. exactly one of the three cyclists takes part;
4. either one or two of the three cyclists take part.

2 Ivan throws three fair dice.

1. List all the possible scores on the three dice which give a total score of 5 , and hence show that the probability of Ivan obtaining a total score of 5 is \(\frac { 1 } { 36 }\).
2. Find the probability of Ivan obtaining a total score of 7.

8 Ann, Bill, Chris and Dipak play a game with a fair cubical die. Starting with Ann they take turns, in alphabetical order, to throw the die. This process is repeated as many times as necessary until a player throws a 6 . When this happens, the game stops and this player is the winner. Find the probability that

1. Chris wins on his first throw,
2. Dipak wins on his second throw,
3. Ann gets a third throw,
4. Bill throws the die exactly three times.

There are 125 sixth-form students in a college, of whom 60 are studying only arts subjects, 40 only science subjects and the rest a mixture of both. Three students are selected at random, without replacement. Find the probability that

1. all three students are studying only arts subjects, [4]
2. exactly one of the three students is studying only science subjects. [3]

A and B are mutually exclusive events. Which one of the following statements must be correct? Tick (\(\checkmark\)) one box. [1 mark] \(P(A \cup B) = P(A) \times P(B)\) \(P(A \cup B) = P(A) - P(B)\) \(P(A \cap B) = 0\) \(P(A \cap B) = 1\)

Events \(A\) and \(B\) are such that \(\text{P}(A) = 0.3\), \(\text{P}(B) = 0.8\) and \(\text{P}(A \text{ and } B) = 0.4\). State, giving a reason in each case, whether events \(A\) and \(B\) are

1. independent, [2]
2. mutually exclusive. [2]

1. A fair die has six faces numbered \(1,2,2,3,3\) and 3 . The die is rolled twice and the number showing on the uppermost face is recorded each time.

Find the probability that the sum of the two numbers recorded is at least 5 .
(5)

4 Two identical biased triangular spinners with sides marked 1,2 and 3 are spun. For each spinner, the probabilities of landing on the sides marked 1,2 and 3 are \(p , q\) and \(r\) respectively. The score is the sum of the numbers on the sides on which the spinners land. You are given that \(\mathrm { P } (\) score is \(6 ) = \frac { 1 } { 36 }\) and \(\mathrm { P } (\) score is \(5 ) = \frac { 1 } { 9 }\). Find the values of \(p , q\) and \(r\).

The Venn diagram shows the numbers of students studying various subjects, in a year group of 100 students. \includegraphics{figure_7} A student is chosen at random from the 100 students. Then another student is chosen from the remaining students. Find the probability that the first student studies History and the second student studies Geography but not Psychology. [4]

4 Single cards, chosen at random, are given away with bars of chocolate. Each card shows a picture of one of 20 different football players. Richard needs just one picture to complete his collection. He buys 5 bars of chocolate and looks at all the pictures. Find the probability that

1. Richard does not complete his collection,
2. he has the required picture exactly once,
3. he completes his collection with the third picture he looks at.

1 Marie wants to choose one student at random from Anthea, Bill and Charlie. She throws two fair coins. If both coins show tails she will choose Anthea. If both coins show heads she will choose Bill. If the coins show one of each she will choose Charlie.

1. Explain why this is not a fair method for choosing the student.
2. Describe how Marie could use the two coins to give a fair method for choosing the student.

2 The 12 houses on one side of a street are numbered with even numbers starting at 2 and going up to 24 . A free newspaper is delivered on Monday to 3 different houses chosen at random from these 12. Find the probability that at least 2 of these newspapers are delivered to houses with numbers greater than 14.

1. The Venn diagram shows the numbers of students studying various subjects, in a year group of 100 students. \includegraphics[max width=\textwidth, alt={}, center]{a65400d1-fadc-4bc7-ba4b-af2df57e390a-04_551_894_395_169} A student is chosen at random from the 100 students. Then another student is chosen from the remaining students. Find the probability that the first student studies History and the second student studies Geography but not Psychology.
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