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Flashcards · Maths

Probability

80 cards from 5 lessons

  1. What is the formula for probability when outcomes are equally likely?

    Number of favourable outcomes divided by the total number of outcomes.

  2. What is the probability of an impossible event?

    0.

  3. What is the probability of a certain event?

    1.

  4. What is \(P(\text{not } A)\)?

    \(1 - P(A)\).

  5. What do all the probabilities in a situation add up to?

    1.

  6. What does mutually exclusive mean?

    The events cannot happen at the same time.

  7. What is the addition rule for mutually exclusive events?

    \(P(A \text{ or } B) = P(A) + P(B)\).

  8. What is the formula for relative frequency?

    The number of times an event happened divided by the total number of trials.

  9. How can you tell if a dice might be biased?

    Its relative frequencies are far from the expected \(\dfrac{1}{6}\), especially with many throws.

  10. How do you find an expected number of outcomes?

    Multiply the probability by the number of trials.

  11. How many outcomes are there for two dice?

    36.

  12. What is the probability of a total of 7 with two dice?

    \(\dfrac{1}{6}\).

  13. What are the outcomes for two coins in order?

    HH, HT, TH and TT.

  14. What is the general addition rule (Higher tier)?

    \(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\).

  15. What is the probability of a heart or a king from a pack of cards (Higher tier)?

    \(\dfrac{16}{52} = \dfrac{4}{13}\).

  16. Why does a larger experiment give a better estimate?

    Relative frequency gets closer to the true probability with more trials.

  17. What does a tree diagram show?

    All the possible outcomes of events happening one after another, with their probabilities.

  18. What do you do along a path in a tree diagram?

    Multiply the probabilities.

  19. What do you do when more than one path gives the result you want?

    Add the path probabilities.

  20. What should the branches leaving one point add up to?

    1.

  21. What does independent mean for two events?

    The first event does not change the probability of the second.

  22. What is different about the branches for events without replacement?

    The second probabilities depend on the first pick.

  23. What is \(P(\text{at least one})\) in terms of \(P(\text{none})\)?

    \(1 - P(\text{none})\).

  24. Rain has probability 0.3 on each of two days. What is the probability of rain on both days?

    \(0.3 \times 0.3 = 0.09\).

  25. With 3 red and 2 blue counters, what is \(P(\text{two reds})\) without replacement?

    \(\dfrac{3}{5} \times \dfrac{2}{4} = \dfrac{3}{10}\).

  26. With 3 red and 2 blue counters, what is \(P(\text{different colours})\) without replacement?

    \(\dfrac{3}{5}\).

  27. What is the probability of any particular outcome of three coin tosses?

    \(\dfrac{1}{8}\).

  28. How do you check a completed tree diagram?

    The outcome probabilities should add up to 1.

  29. If one branch has probability 0.6, what is the other branch from the same point?

    0.4.

  30. Why does the denominator drop to 4 on the second pick from 5 counters?

    One counter has been taken and not replaced.

  31. What does "with replacement" mean?

    The item is put back, so the probabilities stay the same.

  32. Why is "at least one" often quicker with the opposite?

    Only one path (none) needs calculating instead of several.

  33. What does the symbol \(\xi\) represent?

    The universal set, everything being considered.

  34. What does \(A \cap B\) mean?

    The overlap: items in both \(A\) and \(B\).

  35. What does \(A \cup B\) mean?

    Items in \(A\) or \(B\) or both.

  36. What does \(A'\) mean?

    Items not in \(A\), the complement.

  37. What does \(3 \in A\) mean?

    3 is a member of set \(A\).

  38. Where do you start when filling in a Venn diagram?

    In the overlap.

  39. How do you find "football only"?

    Subtract the number who play both from the number who play football.

  40. How do you find the number outside both circles?

    Subtract everyone inside the circles from the total.

  41. How do you find the probability of a region of a Venn diagram?

    The number in the region divided by the total in the rectangle.

  42. What is \(n(A \cup B)\) in terms of \(n(A)\), \(n(B)\) and \(n(A \cap B)\) (Higher tier)?

    \(n(A) + n(B) - n(A \cap B)\).

  43. If \(A = \{2, 4, 6, 8, 10\}\) and \(B = \{3, 6, 9\}\), what is \(A \cap B\)?

    \(\{6\}\).

  44. What is the union of \(A\) and \(B\) above?

    \(\{2, 3, 4, 6, 8, 9, 10\}\).

  45. What does the whole rectangle in a Venn diagram show?

    Everything in the universal set.

  46. What is the notation for the region in \(A\) but not in \(B\) (Higher tier)?

    \(A \cap B'\).

  47. Why is it wrong to add 18 and 14 to find the number playing at least one sport?

    The students who play both are counted twice.

  48. 22 play football, 19 play tennis, 5 play neither out of 40. How many play both?

    \(22 + 19 - 35 = 6\).

  49. What is a two-way table?

    A table that sorts data by two features at once, with totals for rows and columns.

  50. What do the numbers in each row of a two-way table add up to?

    The total at the end of the row.

  51. How do you find a missing number in a two-way table?

    Subtract the known numbers from the row or column total.

  52. What is the grand total in a two-way table?

    The total of all the items, in the bottom right corner.

  53. What is a frequency tree?

    A tree diagram that shows numbers of items rather than probabilities.

  54. What should a pair of branches in a frequency tree add up to?

    The number they came from.

  55. What is 60% of 200?

    120.

  56. How do you find the probability from a two-way table?

    The number in the cell, or cells, divided by the grand total.

  57. How do you estimate the expected number from a probability?

    Multiply the probability by the number in the group.

  58. 13 of 100 students cycle. How many of 500 would you expect to cycle?

    \(\dfrac{13}{100} \times 500 = 65\).

  59. 24 of 100 pupils are girls who walk. What is the probability?

    \(\dfrac{24}{100} = \dfrac{6}{25}\).

  60. How can you check that a two-way table is complete?

    The row totals and the column totals should both add to the grand total.

  61. If 22 of 50 people own a cat only and 14 own a dog only, with 6 owning both, how many have neither?

    \(50 - 42 = 8\).

  62. Why must you check which row or column a number is in?

    Numbers in the wrong row give wrong totals and wrong probabilities.

  63. What is 25% of 120?

    30.

  64. Why draw a table or a tree for a wordy problem?

    It organises the information and shows which numbers are missing.

  65. What is a conditional probability?

    The probability of an event given that something else has happened or is known.

  66. What does \(P(A \mid B)\) mean?

    The probability of \(A\) given \(B\).

  67. What does "given that" do to the group you look at?

    It makes it smaller, to just those that satisfy the condition.

  68. What is the bottom of a conditional probability fraction?

    The number in the given group.

  69. Out of 14 tennis players, 6 also play football. What is \(P(F \mid T)\)?

    \(\dfrac{6}{14} = \dfrac{3}{7}\).

  70. Is \(P(A \mid B)\) always equal to \(P(B \mid A)\)?

    No, they use different groups.

  71. What is the formula for \(P(A \text{ and } B)\) using conditional probability (Higher tier)?

    \(P(A \mid B) \times P(B)\).

  72. How do you rearrange it to find \(P(A \mid B)\) (Higher tier)?

    \(\dfrac{P(A \text{ and } B)}{P(B)}\).

  73. What is the condition for two events to be independent?

    \(P(A \mid B) = P(A)\), or \(P(A \text{ and } B) = P(A) \times P(B)\).

  74. What are the second-stage probabilities on a tree without replacement?

    Conditional probabilities.

  75. Of 50 people who drive, 20 are female. What is the probability that a driver is male?

    \(\dfrac{30}{50} = \dfrac{3}{5}\).

  76. A person who drives is chosen. Why is the denominator 50 and not 100?

    Only the drivers are in the group.

  77. In a tree with 3 red and 2 blue, what is \(P(\text{second red} \mid \text{first red})\)?

    \(\dfrac{2}{4} = \dfrac{1}{2}\).

  78. If \(P(A) = 0.5\), \(P(B) = 0.4\) and \(P(A \text{ and } B) = 0.2\), are they independent?

    Yes, because \(0.5 \times 0.4 = 0.2\).

  79. How do you find the total of the group when you are told the second pick was red?

    Add all the paths that end in red.

  80. Which tools can you use to find conditional probabilities?

    Venn diagrams, two-way tables and tree diagrams.