Probability · Class 10 Mathematics: 44 flashcards
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1. State the classical definition of probability.
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$P(E) = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$.
2. What condition must the outcomes satisfy?
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They must be equally likely.
3. Give an example where the definition does not apply.
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A drawing pin tossed in the air: it lands point-up or point-down, but not equally often.
4. How many outcomes does tossing two coins have?
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Four: HH, HT, TH, TT.
5. Find the probability of exactly one head when two coins are tossed.
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$\frac24 = \frac12$, since HT and TH both qualify.
6. What is the difference between an outcome and an event?
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An outcome is a single result; an event is a collection of outcomes.
7. Find the probability of an even number on one die.
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$\frac36 = \frac12$ — the outcomes 2, 4 and 6.
8. Why must every probability lie between 0 and 1?
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The favourable count cannot be negative nor exceed the total.
9. What does $P(E) = 0$ mean?
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The event is impossible — no outcome is favourable.
10. What does $P(E) = 1$ mean?
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The event is certain — every outcome is favourable.
11. What is the probability of rolling a 7 on one ordinary die?
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0 — no face shows 7.
12. What is the complementary event $\bar{E}$?
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The event that $E$ does not happen.
13. State the complement rule.
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$P(\bar{E}) = 1 - P(E)$.
14. The probability of drawing a king is $\frac{1}{13}$. Find the probability of not drawing one.
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$1 - \frac{1}{13} = \frac{12}{13}$.
15. Two dice are thrown. Find the probability that the total is not 7.
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$1 - \frac{6}{36} = \frac56$.
16. When is the complement worth using?
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When the event asked about is harder to count than its opposite — often signalled by 'not', 'at least' or 'none'.
17. What do the probabilities of all the outcomes add to?
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Exactly 1.
18. In what forms may a probability be written?
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As a fraction, a decimal or a percentage — but never as a ratio such as 1 : 6.
19. Should $\frac{18}{36}$ be left as it is?
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No — reduce it to $\frac12$.
20. What is the difference between 'at least one head' and 'exactly one head'?
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'At least one' includes the case of two heads; 'exactly one' excludes it.
21. Is a very small probability the same as zero?
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No — only an event that cannot occur at all has probability 0.
22. What is the first step in any probability question?
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List the outcomes.
23. How many outcomes do $n$ coins have?
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$2^n$ — each coin doubles the count.
24. List the outcomes for three coins.
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HHH, HHT, HTH, HTT, THH, THT, TTH, TTT — eight in all.
25. With three coins, find $P(\text{exactly two heads})$.
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$\frac38$ — HHT, HTH and THH.
26. With three coins, find $P(\text{at least one head})$.
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$1 - \frac18 = \frac78$, since only TTT has no head.
27. Are the four head-counts 0, 1, 2, 3 equally likely for three coins?
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No — one head happens three ways, three heads only one.
28. How many outcomes do two dice have?
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36, since $6 \times 6$.
29. Are $(2,5)$ and $(5,2)$ the same outcome?
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No — the dice are distinguishable, so they are different outcomes.
30. How many ways can two dice total 7?
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Six: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).
31. Find $P(\text{total } 12)$ for two dice.
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$\frac{1}{36}$ — only (6,6).
32. What is a double, and how likely is it?
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Both dice showing the same number; 6 outcomes, so $\frac{6}{36} = \frac16$.
33. How many cards are in a pack, and how are they arranged?
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52 cards: four suits of 13.
34. Which suits are red?
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Hearts and diamonds — 26 red cards in all.
35. Which cards are the face cards?
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Jack, queen and king: 3 per suit, 12 in the pack.
36. Is an ace a face card?
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No. There are 12 face cards and 4 aces, making 16 cards that are one or the other.
37. Find $P(\text{a heart})$ from a full pack.
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$\frac{13}{52} = \frac14$.
38. Find $P(\text{a king})$ from a full pack.
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$\frac{4}{52} = \frac{1}{13}$.
39. Find $P(\text{a red king})$.
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$\frac{2}{52} = \frac{1}{26}$ — the kings of hearts and diamonds.
40. Find $P(\text{a face card})$.
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$\frac{12}{52} = \frac{3}{13}$.
41. How many cards are a king or a heart?
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16, not 17 — the king of hearts must not be counted twice.
42. All face cards are removed. How many cards remain, and how many are hearts?
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40 remain, of which 10 are hearts.
43. Find $P(\text{a heart})$ after the face cards are removed.
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$\frac{10}{40} = \frac14$.
44. What should you write down first in any question of this topic?
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The total number of outcomes, adjusted for any removals.
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