Given conditional, find joint or marginal

5 Eric has three coins. One of the coins is fair. The other two coins are each biased so that the probability of obtaining a head on any throw is \(\frac { 1 } { 4 }\), independently of all other throws. Eric throws all three coins at the same time. Events \(A\) and \(B\) are defined as follows.
\(A\) : all three coins show the same result
\(B\) : at least one of the biased coins shows a head

1. Show that \(\mathrm { P } ( B ) = \frac { 7 } { 16 }\).
2. Find \(\mathrm { P } ( A \mid B )\).
   The random variable \(X\) is the number of heads obtained when Eric throws the three coins.
3. Draw up the probability distribution table for \(X\).

3 A shop sells two makes of coffee, Café Premium and Café Standard. Both coffees come in two sizes, large jars and small jars. Of the jars on sale, \(65 \%\) are Café Premium and \(35 \%\) are Café Standard. Of the Café Premium, 40\% of the jars are large and of the Café Standard, 25\% of the jars are large. A jar is chosen at random.

1. Find the probability that the jar is small.
2. Find the probability that the jar is Café Standard given that it is large.

4 A supermarket has a large stock of eggs. 40\% of the stock are from a firm called Eggzact. 12\% of the stock are brown eggs from Eggzact. An egg is chosen at random from the stock. Calculate the probability that

1. this egg is brown, given that it is from Eggzact,
2. this egg is from Eggzact and is not brown.

2 Isobel plays football for a local team. Sometimes her parents attend matches to watch her play.

• \(A\) is the event that Isobel's parents watch a match.
• \(B\) is the event that Isobel scores in a match.

You are given that \(\mathrm { P } ( B \mid A ) = \frac { 3 } { 7 }\) and \(\mathrm { P } ( A ) = \frac { 7 } { 10 }\).

1. Calculate \(\mathrm { P } ( A \cap B )\). The probability that Isobel does not score and her parents do not attend is 0.1 .
2. Draw a Venn diagram showing the events \(A\) and \(B\), and mark in the probability corresponding to each of the regions of your diagram.
3. Are events \(A\) and \(B\) independent? Give a reason for your answer.
4. By comparing \(\mathrm { P } ( B \mid A )\) with \(\mathrm { P } ( B )\), explain why Isobel should ask her parents not to attend.

2 Each day the probability that Ashwin wears a tie is 0.2 . The probability that he wears a jacket is 0.4 . If he wears a jacket, the probability that he wears a tie is 0.3 .

1. Find the probability that, on a randomly selected day, Ashwin wears a jacket and a tie.
2. Draw a Venn diagram, using one circle for the event 'wears a jacket' and one circle for the event 'wears a tie'. Your diagram should include the probability for each region.
3. Using your Venn diagram, or otherwise, find the probability that, on a randomly selected day, Ashwin
   (A) wears either a jacket or a tie (or both),
   (B) wears no tie or no jacket (or wears neither).

3 Each weekday Alan drives to work. On his journey, he goes over a level crossing. Sometimes he has to wait at the level crossing for a train to pass.

• \(W\) is the event that Alan has to wait at the level crossing.
• \(L\) is the event that Alan is late for work.

You are given that \(\mathrm { P } ( L \mid W ) = 0.4 , \mathrm { P } ( W ) = 0.07\) and \(\mathrm { P } ( L \cup W ) = 0.08\).

1. Calculate \(\mathrm { P } ( L \cap W )\).
2. Draw a Venn diagram, showing the events \(L\) and \(W\). Fill in the probability corresponding to each of the four regions of your diagram.
3. Determine whether the events \(L\) and \(W\) are independent, explaining your method clearly.

8 For the events \(L\) and \(M , \mathrm { P } ( L \mid M ) = 0.2 , \mathrm { P } ( M \mid L ) = 0.4\) and \(\mathrm { P } ( M ) = 0.6\).

1. Find \(\mathrm { P } ( L )\) and \(\mathrm { P } \left( L ^ { \prime } \cup M ^ { \prime } \right)\).
2. Given that, for the event \(N , \mathrm { P } ( N \mid ( L \cap M ) ) = 0.3\), find \(\mathrm { P } \left( L ^ { \prime } \cup M ^ { \prime } \cup N ^ { \prime } \right)\).

1. Events \(A\) and \(B\) are such that

$$\mathrm { P } ( A ) = 0.5 \quad \mathrm { P } ( A \mid B ) = \frac { 2 } { 3 } \quad \mathrm { P } \left( A ^ { \prime } \cup B ^ { \prime } \right) = 0.6$$

1. Find \(\mathrm { P } ( B )\)
2. Find \(\mathrm { P } \left( A ^ { \prime } \mid B ^ { \prime } \right)\) The event \(C\) has \(\mathrm { P } ( C ) = 0.15\) The events \(A\) and \(C\) are mutually exclusive. The events \(B\) and \(C\) are independent.
3. Find \(\mathrm { P } ( B \cap C )\)
4. Draw a Venn diagram to illustrate the events \(A , B\) and \(C\) and the probabilities for each region.
   

2. A group of office workers were questioned for a health magazine and \(\frac { 2 } { 5 }\) were found to take regular exercise. When questioned about their eating habits \(\frac { 2 } { 3 }\) said they always eat breakfast and, of those who always eat breakfast \(\frac { 9 } { 25 }\) also took regular exercise. Find the probability that a randomly selected member of the group

1. always eats breakfast and takes regular exercise,
2. does not always eat breakfast and does not take regular exercise.
3. Determine, giving your reason, whether or not always eating breakfast and taking regular exercise are statistically independent.

1. \(\quad A\) and \(B\) are two events such that

$$\mathrm { P } ( B ) = \frac { 1 } { 2 } \quad \mathrm { P } ( A \mid B ) = \frac { 2 } { 5 } \quad \mathrm { P } ( A \cup B ) = \frac { 13 } { 20 }$$

1. Find \(\mathrm { P } ( A \cap B )\).
2. Draw a Venn diagram to show the events \(A , B\) and all the associated probabilities. Find
3. \(\mathrm { P } ( A )\)
4. \(\mathrm { P } ( B \mid A )\)
5. \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\)

8. For the events \(A\) and \(B\), $$\mathrm { P } \left( A ^ { \prime } \cap B \right) = 0.22 \text { and } \mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right) = 0.18$$

1. Find \(\mathrm { P } ( A )\).
2. Find \(\mathrm { P } ( A \cup B )\). Given that \(\mathrm { P } ( A \mid B ) = 0.6\)
3. find \(\mathrm { P } ( A \cap B )\).
4. Determine whether or not \(A\) and \(B\) are independent.

6. For any married couple who are members of a tennis club, the probability that the husband has a degree is \(\frac { 3 } { 5 }\) and the probability that the wife has a degree is \(\frac { 1 } { 2 }\). The probability that the husband has a degree, given that the wife has a degree, is \(\frac { 11 } { 12 }\). A married couple is chosen at random.

1. Show that the probability that both of them have degrees is \(\frac { 11 } { 24 }\).
2. Draw a Venn diagram to represent these data. Find the probability that
3. only one of them has a degree,
4. neither of them has a degree. Two married couples are chosen at random.
5. Find the probability that only one of the two husbands and only one of the two wives have degrees.

3. The events \(A\) and \(B\) are such that $$\mathrm { P } ( A ) = \frac { 7 } { 12 } , \mathrm { P } ( A \cap B ) = \frac { 1 } { 4 } \quad \text { and } \mathrm { P } ( A \mid B ) = \frac { 2 } { 3 }$$ Find

1. \(\mathrm { P } ( B )\),
2. \(\mathrm { P } ( A \cup B )\),
3. \(\mathrm { P } \left( B \mid A ^ { \prime } \right)\).

2. Events \(A\) and \(B\) are such that \(P ( A \cup B ) = 0.95 , P ( A \cap B ) = 0.6\) and \(P ( A \mid B ) = 0.75\).
i. Find \(P ( B )\).
ii. Find \(P ( A )\).
iii. Show that the events \(A ^ { \prime }\) and \(B\) are independent.
[0pt] [BLANK PAGE]