Independent and Dependent Events: Probability Calculations

2. 1. Decide whether these events are independent or dependent. You have a jar with 24 chocolates and 14 jellies. You take one sweetatrandomfrom the jar, eatit, and thentake a second sweetatrandomfrom the jar. (i) Dependent: The sweetis not replacedsothere are fewersweets to choosefrom on the nextpick. Derek has a blue, a red, and a green tie. He also has a blue and a green shirt. He chooses a randomtie and shirt for worktoday. (ii) Independent: The choice of tiedoes not affect the choice of shirts.

3. 1. Decide whether these events are independent or dependent. Amy playscardgames. Shepicks a cardatrandom. Thenwithout putting the first card back, shepicks a second cardatrandom. (iii) Dependent: The cardis not replacedsothere are fewercards to choosefrom on the nextpick. Ian has 14 coins. He takesthree of thematrandom, thenheputsthese back, and thenpickstwo more coins atrandom. (iv) Independent: The coins are replacedsothere are the samenumbers of coins to pickfromeach time.

4. 1. Decide whether these events are independent or dependent. Jeff has threechildren. His first twochildren are boys. His last childis a girl. (v) Independent: Eachchild’sgenderis not affected by the gender of the others. A plant has four redflowers and twoblueflowers. Brenda picks one flowerfrom the plant. Aftersome time, hersisterpicks a flowerfrom the same plant. (vi) Dependent: The flowersaren’treplaced, sothere are fewerflowers to choosefrom the nextpick.

5. 2. You roll a fair six-sided die twice. What is the probability the first roll shows a five and the second roll shows a six? P(5 on 1st Roll and 6 on 2nd Roll) P(5 and 6) ‘and’ means multiply = P(5) × P(6)

6. 3. There are nine mugs in Tom’s cupboard, four blue and five green. He randomly selects one to use on Monday, washes it and replaces it. He randomly selects another on Tuesday. What is the probability that: he uses a bluemugbothdays? (i) There are 4 blue mugs and 5 green mugs giving a total of 9 mugs. P(Blue on Monday and Blue on Tuesday) P(Blue and Blue) = P(Blue) × P(Blue)

7. 3. You There are nine mugs in Tom’s cupboard, four blue and five green. He randomly selects one to use on Monday, washes it and replaces it. He randomly selects another on Tuesday. What is the probability that: he uses a bluemug on Monday and a green mug on Tuesday? (ii) P(Blue on Monday and Green on Tuesday) P(Blue and Green) = P(Blue) × P(Green)

8. 4. Shannon spins a spinner numbered 1 to 7, and tosses a coin. What is the probability she gets: an oddnumber on the spinner? (i) P(Odd)

9. 4. Shannon spins a spinner numbered 1 to 7, and tosses a coin. What is the probability she gets: an oddnumber on the spinner and a tail on the tail? (ii) P(Odd and Tail) = P(Odd) × P(Tail)

10. 5. You flip a coin and then roll a fair six-sided die. Draw a treediagram to representthis situation. (i) Write the probabilities on each branch.

11. 5. You flip a coin and then roll a fair six-sided die. Use the tree diagram to calculate the probability that: the coin lands heads-up and the die shows a three? (ii) P(Head and 3) = P(Head) × P(3)

12. 5. You flip a coin and then roll a fair six-sided die. Use the tree diagram to calculate the probability that: the coin lands tails-up and the die shows a numbergreaterthan four? (iii) P(Tail and 5, 6) = P(Tail) × P(5 or 6)

13. 5. You flip a coin and then roll a fair six-sided die. Use the tree diagram to calculate the probability that: the coin lands heads-up and the die shows an evennumber? (iv) P(Head and 2, 4, 6) = P(Head) × P(2 or 4 or 6)

14. 5. You flip a coin and then roll a fair six-sided die. Use the tree diagram to calculate the probability that: the coin lands heads-up and the die shows a prime number? (v) P(Head and 2, 3, 5) = P(Head) × P(2 or 3 or 5)

15. 6. The names of 9 boys and 11 girls from your class are put into a hat. Two names are selected without replacement. Draw a treediagram to representthis situation. (i) Write the probabilities on each branch.

16. 6. The names of 9 boys and 11 girls from your class are put into a hat. Two names are selected without replacement. What is the probability that: the twonameschosenwillbothbe girls? (ii) There are 9 boys and 11 girls sothere are 20 students in total. P(1st is a girl and 2nd is a girl) = P(Girl) × P(Girl) 1 girl gone 1 person gone

17. 6. The names of 9 boys and 11 girls from your class are put into a hat. Two names are selected without replacement. What is the probability that: the first namewillbe a boy and the second willbe a girl? (iii) P(1st is a boy and 2nd is a girl) = P(Boys) × P(Girls) 1 person gone

18. 6. The names of 9 boys and 11 girls from your class are put into a hat. Two names are selected without replacement. What is the probability that: bothnameswillbe boys? (iv) P(1st is a boy and 2nd is a boy) = P(Boys) × P(Boys) 1 boy gone 1 person gone

19. 7. A card is chosen at random from a standard deck of 52 playing cards. Without replacing it, a second card is chosen. What is the probability that the first card chosen is a queen and the second card chosen is a jack? P(1st is a Queen and 2nd is a Jack) = P(Queen) × P(Jack) 1 card gone

20. 8. A bowl of fruit is on the table. It contains four apples, six oranges and three bananas. Alan and Kenneth come home from school and randomly grab one fruit each. What is the probability that both grab apples? There are 4 apples, 6 oranges and 3 bananas sothere are 13 pieces of fruit in total. P(Apple and Apple) = P(Apple) × P(Apple)

21. 9. Your drawer contains 11 red socks and six blue socks. It’s too dark to see which are which, but you grab two anyway. Draw a treediagram to representthis information. Write the probabilities on eachbranch. (i)

22. 9. Your drawer contains 11 red socks and six blue socks. It’s too dark to see which are which, but you grab two anyway. What is the probability that: the first sockisred and the second isblue? (ii) P(1st is Red and 2nd is Blue) = P(Red) × P(Blue)

23. 9. Your drawer contains 11 red socks and six blue socks. It’s too dark to see which are which, but you grab two anyway. What is the probability that: the first sockisblue and the second isred? (iii) P(1st is Blue and 2nd is Red) = P(Blue) × P(Red)

24. 9. Your drawer contains 11 red socks and six blue socks. It’s too dark to see which are which, but you grab two anyway. What is the probability that: bothsocks are blue? (iv) P(Blue and Blue) = P(Blue) × P(Blue)

25. 9. Your drawer contains 11 red socks and six blue socks. It’s too dark to see which are which, but you grab two anyway. What is the probability that: bothsocks are red? (v) P(Red and Red) = P(Red) × P(Red)

26. 10. The game of backgammon uses two standard dice, each with the numbers one through six. You need to roll double twos to win the game. What is the probability you will get that result on your next roll? P(2 and 2) = P(2) × P(2)

27. 11. I roll three dice. Are theseeventsdependent or independent? (i) Theseevents are independentbecause the result of rollingeach die does not influence the next die.

28. 11. I roll three dice. Whatis the probabilitythat I getthreeones? (ii) P(1 and 1 and 1) = P(1) × P(1) × P(1)

29. 12. I draw a card from a standard deck, replace it, and draw another. Are theseeventsdependent or independent? (i) Theseevents are independentbecause the cardisreplacedsothere are the samenumber of cards to pickfrom on the second draw.

30. 12. I draw a card from a standard deck, replace it, and draw another. Whatis the probability I gettwo aces? (ii) P(Ace and Ace) = P(Ace) × P(Ace)

31. 13. Four cards are chosen at random from a deck of 52 cards, one after the other, without replacement. Are theseeventsdependent or independent? (i) Theseevents are dependentbecause the cards are not replacedsothere are fewercards to pickfromeach time.

32. 13. Four cards are chosen at random from a deck of 52 cards, one after the other, without replacement. Whatis the probability of choosing a ten, a nine, an eight and a seven in thatorder? (ii) P(10 and 9 and 8 and 7) = P(10) × P(9) × P(8) × P(7)

33. 14. Henry has three black shirts and seven blue shirts in his wardrobe. Two shirts are drawn without replacement from the wardrobe. What is the probability that both of the shirts are black? There are 3 black shirts and 7 blueshirtsso a total of 10 shirts. P(Black and black) = P(Black) × P(Black)

34. 15. An archer always hits a circular target with each arrow shot. He hits the bullseye two out of every five shots on average. If he takes three shots at the target, calculate the probability that he hits the bullseye: every time (i) P(Bullseye and Bullseye and Bullseye) Every time : = P(Bullseye) × P(Bullseye) × P(Bullseye)

35. 15. An archer always hits a circular target with each arrow shot. He hits the bullseye two out of every five shots on average. If he takes three shots at the target, calculate the probability that he hits the bullseye: the first two times but not the third (ii) P(Bullseye and Bullseye and no Bullseye) = P(Bullseye) × P(Bullseye) × P(No Bullseye) P(No Bullseye) = 1 – P(Bullseye)

36. 15. An archer always hits a circular target with each arrow shot. He hits the bullseye two out of every five shots on average. If he takes three shots at the target, calculate the probability that he hits the bullseye: the first two times but not the third (ii) P(Bullseye and Bullseye and no Bullseye) = P(Bullseye) × P(Bullseye) × P(No Bullseye) P(Bullseye and Bullseye and no Bullseye)

37. 15. An archer always hits a circular target with each arrow shot. He hits the bullseye two out of every five shots on average. If he takes three shots at the target, calculate the probability that he hits the bullseye: on no occasion. (iii) P(No Bullseye and no Bullseye and no Bullseye) = P(No Bullseye) × (no Bullseye) × (no Bullseye)

38. 16. A class has 14 boys and 16 girls. Three students are chosen at random to represent the class at a school event. What is the probability that: they are all girls? (i) There are 14 boys and 16 girls so 30 students in total. P(all girls) = P(Girl) × P(Girl) × P(Girl) (Dependent events)

39. 16. A class has 14 boys and 16 girls. Three students are chosen at random to represent the class at a school event. What is the probability that: they are all boys? (ii) P(all boys) = P(Boy) × P(Boy) × P(Boy)

40. 17. Emma says that it is very difficult to get five heads in a row when you flip a coin. Brian disagrees and says he has done it loads of times. Whichperson do youagreewith? (i) I agreewith Emma as the probabilityislow of this happening.

41. 17. Emma says that it is very difficult to get five heads in a row when you flip a coin. Brian disagrees and says he has done it loads of times. Neither of them have a coin to try it so they decide to work the chances out mathematically. Show the calculationstheywould use to do this. (ii) P(Head and Head and Head and Head and Head) 5 heads in a row : = P(Head) × P(Head) × P(Head) × P(Head) × P(Head)

42. 17. Emma says that it is very difficult to get five heads in a row when you flip a coin. Brian disagrees and says he has done it loads of times. Neither of them have a coin to try it so they decide to work the chances out mathematically. Do youwant to change youranswer to (i) above? Justifyyouranswerusing the calculations in (ii) above. (iii) The probability of this happening isonlywhichmeans once in 32 attempts.