Probability Definitions

10 The probabilities of events \(A , B\) and \(C\) are related, as shown in the Venn diagram below.
\(\varepsilon\)
\includegraphics[max width=\textwidth, alt={}, center]{076ea8e9-9295-46d2-b5f9-b27fa969129e-15_620_1200_799_443} Find the value of \(x\). Circle your answer.
\(0.11 \quad 0.46 \quad 0.54 \quad 0.89\)

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.

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.

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.

4. The events \(A\) and \(B\) are such that $$\mathrm { P } ( A ) = 0.5 , \mathrm { P } ( B ) = 0.42 \text { and } \mathrm { P } ( A \cup B ) = 0.76$$ Find

1. \(\mathrm { P } ( A \cap B )\),
2. \(\quad \mathrm { P } \left( A ^ { \prime } \cup B \right)\),
3. \(\mathrm { P } \left( B \mid A ^ { \prime } \right)\).
4. Show that events \(A\) and \(B\) are not independent.

3 The probability that Chipping FC win a league football match is \(\mathrm { P } ( W ) = 0.4\).

1. Calculate the probability that Chipping FC fail to win each of their next two league football matches. The probability that Chipping FC lose a league football match is \(\mathrm { P } ( L ) = 0.3\).
2. Explain why \(\mathrm { P } ( W ) + \mathrm { P } ( L ) \neq 1\).

7 You are given that \(P ( A ) = 0.6 , P ( B ) = 0.5\) and \(P ( A \cup B ) ^ { \prime } = 0.2\).

1. Find \(\mathrm { P } ( \mathrm { A } \cap \mathrm { B } )\).
2. Find \(\mathrm { P } ( \mathrm { A } \mid \mathrm { B } )\).
3. State, with a reason, whether \(A\) and \(B\) 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.

4 Two fair dice are thrown.

1. Event \(A\) is 'the scores differ by 3 or more'. Find the probability of event \(A\).
2. Event \(B\) is 'the product of the scores is greater than 8 '. Find the probability of event \(B\).
3. State with a reason whether events \(A\) and \(B\) are mutually exclusive.

18

1. Bag A contains 7 blue discs, 4 red discs and 1 yellow disc. Two discs are drawn at random from bag A without replacement.
   Find the probability that exactly one of the discs is blue.
   18
2. Bag A contains 7 blue discs, 4 red discs and 1 yellow disc.

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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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 \(r\).

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.

5. Labrador puppies may be black, yellow or chocolate in colour. Some information about a litter of 9 puppies is given in the table. Four puppies are chosen at random to train as guide dogs.
(b) Determine the probability that at least 3 black puppies are chosen.
(c) Determine the probability that exactly 3 females are chosen given that at least 3 black puppies are chosen.
(d) Explain whether the 2 events
'choosing exactly 3 females' and 'choosing at least 3 black puppies' are independent events. A firm claims that no more than \(2 \%\) of their packets of sugar are underweight. A market researcher believes that the actual proportion is greater than \(2 \%\). In order to test the firm's claim, the researcher weighs a random sample of 600 packets and carries out a hypothesis test, at the \(5 \%\) significance level, using the null hypothesis \(p = 0.02\).
(a) Given that the researcher's null hypothesis is correct, determine the probability that the researcher will conclude that the firm's claim is incorrect.
(b) The researcher finds that 18 out of the 600 packets are underweight. A colleague says
" 18 out of 600 is \(3 \%\), so there is evidence that the actual proportion of underweight bags is greater than \(2 \%\)." Criticise this statement.