1. For events and , , and . Find , and . Are and mutually exclusive? 2. For events and , , and Find , and . Are and mutually exclusive? 3. For events and , , and Find . Are and mutually exclusive? Are they independent? 4. For events and , , and Find . Are and mutually exclusive? Are they independent? 5. For events and , , and . Find and . Are and independent? 6. For events and , , and Find , , and 7. , and . Find and 8. , , . Find and 9. For events and , and . Find if: a) and are mutually exclusive b) and are independent c) What is the least possible value of ? 10. Of the married couples in the UK, 60% of the husbands and 30% of the wives have an (extra-marital) lover. For 10% of the couples, both have secret lovers. Find the proportion of couples for which: a) All least one of them has a lover b) Neither has a lover 11. In Winter, the probability that it is too cold to play football is 1/4. The probability that it will be too wet to play is 1/3. 10% of the time, it is both too cold and too wet to play. What is the probability we will be able to play on the 13 th December? 12. In Harrow, 70% of households are Indian and 80% of households have a paved over their garden. a) What is the greatest proportion that are not Indian and have a decent garden? b) If 15% of the households are not Indian and actually have a proper garden, what proportion of households are Indian with concrete for a garden? 13. A canteen serves three-course meals. Everyone has a main course, but only 10% have all 3-courses. The proportions having starter and dessert are 40% and 30% respectively. Rolly is enjoying dessert. What is the probability he had a starter? 14. The proportion of students that read Maths Monthly is 1%. The proportion that read Calculus Quarterly is 0.5%. The probability that a Maths Monthly reader also reads Calculus Quarterly is 0.4. When Miren was caught by his friends reading Calculus Quarterly, they said: “I bet you read Maths Monthly too”. What is the probability their accusation is correct? 15. A student is taking A-levels in Alchemy and Astrology. The probabilities that she will pass these are 0.8 and 0.7 respectively. The chance she will pass both is 0.6 a) If she passes Alchemy, what is the probability that she will pass Astrology? b) If she passes only one of them, what is the probability that is was Alchemy? 16. An drug dealer has clients in three areas, A, B and C, in the ratio 3:2:5. The probabilities that a client from one of these areas will die 0.5, 0.2 and 0.1 respectively. a) Draw a Venn diagram to represent this information fully b) If a client has just died, what is the probability he was a client from area C? 17. In a coffee shop; 70% order a latte, 20% order a cappuccino and 10% order an espresso. Of those ordering a latte, 3/5 take sugar; whereas only 1/4 of those ordering a cappuccino and 11/20 of those ordering an espresso take sugar. a) Draw a Venn diagram to represent this information fully b) Find the chance that a person who doesn’t take sugar will order a cappuccino 18. A criminologist is examining the records of the criminals on his records Of the 200 cases, he finds that committed by 100 are male perpetrators, 40 are for violent crime and in 60 of the cases, the judge awarded a prison sentence Looking more closely, he found that: 12 violent crimes committed were by male perpetrators, 20 males criminals that were given a prison sentence, 24 violent cases that resulted in a prison sentence, 6 cases involved males and violent crime and resulted in a prison sentence. a) Represent this information on a Venn diagram b) What proportion of violent criminals that were imprisoned, were male? 19. Events , and are defined as follows: = attending a private school = receiving extra tuition = getting three or more A/A*-grades at A-level Given: , , , and , , , and What proportion of the students that attend private school and have extra tuition will end up with three A-grades or higher at A-level? 20. All the girls in a stats class support at least one of the top three football teams: 1/3 support Manchester United, 1/4 support Chelsea, 13/24 support the Arsenal A few support two of the teams (I know, but you know what girls are like…) , but not even a girl is crazy enough to support all three! 1/24 support both Man U and Arsenal, 1/30 support both Man U and Chelsea. Tilly is wearing her Chelsea scarf. Find the probability that Tilly supports Arsenal?
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1. For events and , , and .
Find , and . Are and mutually exclusive?
2. For events and ,
,
and
Find , and . Are and mutually exclusive?
3. For events and ,
,
and
Find . Are and mutually exclusive? Are they independent?
4. For events and , , and Find . Are and mutually exclusive? Are they independent?
5. For events and , , and .
Find and . Are and independent?
6. For events and , , and
Find , , and
7.
,
and
. Find and
8.
,
, . Find and
9. For events and ,
and
. Find if:
a) and are mutually exclusive b) and are independent
c)
What is the least possible value of ?
10. Of the married couples in the UK, 60% of the husbands and 30% of the wives have an (extra-marital) lover. For 10% of the couples, both have secret lovers. Find the proportion of couples for which: a) All least one of them has a lover b) Neither has a lover
11. In Winter, the probability that it is too cold to play football is 1/4. The probability that it will be too wet to play is 1/3. 10% of the time, it is both too cold and too wet to play. What is the probability we will be able to play on the 13
th December?
12. In Harrow, 70% of households are Indian and 80% of households have a paved over their garden. a) What is the greatest proportion that are not Indian and have a decent garden? b) If 15% of the households are not Indian and actually have a proper garden, what proportion of households are Indian with concrete for a garden?
13. A canteen serves three-course meals. Everyone has a main course, but only 10% have all 3-courses. The proportions having starter and dessert are 40% and 30% respectively. Rolly is enjoying dessert. What is the probability he had a starter?
14. The proportion of students that read Maths Monthly is 1%. The proportion that read Calculus Quarterly is 0.5%. The probability that a Maths Monthly reader also reads Calculus Quarterly is 0.4. When Miren was caught by his friends reading Calculus Quarterly, they said: “I bet you read Maths Monthly too”. What is the probability their accusation is correct?
15. A student is taking A-levels in Alchemy and Astrology. The probabilities that she will pass these are 0.8 and 0.7 respectively. The chance she will pass both is 0.6 a) If she passes Alchemy, what is the probability that she will pass Astrology? b) If she passes only one of them, what is the probability that is was Alchemy?
16. An drug dealer has clients in three areas, A, B and C, in the ratio 3:2:5. The probabilities that a client from one of these areas will die 0.5, 0.2 and 0.1 respectively. a) Draw a Venn diagram to represent this information fully b) If a client has just died, what is the probability he was a client from area C?
17. In a coffee shop; 70% order a latte, 20% order a cappuccino and 10% order an espresso. Of those ordering a latte, 3/5 take sugar; whereas only 1/4 of those ordering a cappuccino and 11/20 of those ordering an espresso take sugar. a) Draw a Venn diagram to represent this information fully b) Find the chance that a person who doesn’t take sugar will order a cappuccino
18. A criminologist is examining the records of the criminals on his records Of the 200 cases, he finds that committed by 100 are male perpetrators, 40 are for violent crime and in 60 of the cases, the judge awarded a prison sentence Looking more closely, he found that: 12 violent crimes committed were by male perpetrators, 20 males criminals that were given a prison sentence, 24 violent cases that resulted in a prison sentence, 6 cases involved males and violent crime and resulted in a prison sentence. a) Represent this information on a Venn diagram b) What proportion of violent criminals that were imprisoned, were male?
19. Events , and are defined as follows: = attending a private school = receiving extra tuition = getting three or more A/A*-grades at A-level
Given:
,
,
,
and
,
,
, and
What proportion of the students that attend private school and have extra tuition will end up with three A-grades or higher at A-level?
20. All the girls in a stats class support at least one of the top three football teams: 1/3 support Manchester United, 1/4 support Chelsea, 13/24 support the Arsenal A few support two of the teams (I know, but you know what girls are like…), but not even a girl is crazy enough to support all three! 1/24 support both Man U and Arsenal, 1/30 support both Man U and Chelsea. Tilly is wearing her Chelsea scarf. Find the probability that Tilly supports Arsenal?
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