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(ii) Probability is expressed as

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Here are the answers to the multiple-choice questions:

Part ii

Probability is expressed as

Probability can be expressed as a ratio (e.g., 1:2), a proportion (e.g., 0.5), or a percentage (e.g., 50%).
The correct option is (d).

(d) all of the above\boxed{\text{(d) all of the above}}

Part iii

When two events cannot happen simultaneously in a single trial, the events are said to be

Events that cannot occur at the same time are defined as mutually exclusive events.
The correct option is (c).

(c) mutually Exclusive\boxed{\text{(c) mutually Exclusive}}

Part iv

An event consisting of those elements, which are not in A is called

The set of all outcomes in the sample space that are not in event A is known as the complementary event of A.
The correct option is (d).

(d) complementary event\boxed{\text{(d) complementary event}}

Part v

If A is an event, the conditional probability of A given A is equal to

The conditional probability P(A∣A)P(A|A) means the probability of event A occurring, given that event A has already occurred. If A has already occurred, its probability is certain.

P(A∣A)=P(A∩A)P(A)=P(A)P(A)=1P(A|A) = \frac{P(A \cap A)}{P(A)} = \frac{P(A)}{P(A)} = 1

The correct option is (b).

(b) one\boxed{\text{(b) one}}

Part vi· 3 steps

If a coin is tossed three times, then the probability of getting at most one head is equal to

  1. List all possible outcomes when a coin is tossed three times.

    The sample space S={HHH,HHT,HTH,THH,HTT,THT,TTH,TTT}S = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}.
    The total number of outcomes is 8.

  2. Identify outcomes with "at most one head" (meaning 0 heads or 1 head).

    Outcomes with 0 heads: {TTT}\{TTT\} (1 outcome)
    Outcomes with 1 head: {HTT,THT,TTH}\{HTT, THT, TTH\} (3 outcomes)
    Total favorable outcomes = 1+3=41 + 3 = 4.

  3. Calculate the probability.

    P(at most one head)=Number of favorable outcomesTotal number of outcomes=48=12P(\text{at most one head}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{8} = \frac{1}{2}

    The correct option is (c).

    (c) 1/2\boxed{\text{(c) 1/2}}

Part vii· 3 steps

The probability of throwing an even sum with two fair dice is

  1. Determine the total number of outcomes.

    When two fair dice are thrown, each die has 6 possible outcomes. The total number of combined outcomes is 6×6=366 \times 6 = 36.

  2. Determine the number of outcomes that result in an even sum.

    A sum is even if both dice show even numbers (E, E) or both dice show odd numbers (O, O).

    • Even numbers on a die: {2,4,6}\{2, 4, 6\} (3 outcomes)
    • Odd numbers on a die: {1,3,5}\{1, 3, 5\} (3 outcomes) Number of (E, E) outcomes = 3×3=93 \times 3 = 9. Number of (O, O) outcomes = 3×3=93 \times 3 = 9. Total favorable outcomes (even sum) = 9+9=189 + 9 = 18.

  3. Calculate the probability.

    P(even sum)=Number of favorable outcomesTotal number of outcomes=1836=12P(\text{even sum}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{18}{36} = \frac{1}{2}

    The correct option is (d).

    (d) 1/2\boxed{\text{(d) 1/2}}

Part viii

The probability of six on a fair die is 1/6. The probability of not six is

Let AA be the event of rolling a six. P(A)=16P(A) = \frac{1}{6}.
The probability of not rolling a six is the complement of event AA, denoted as P(A′)P(A').

P(A′)=1−P(A)P(A') = 1 - P(A) P(not six)=1−16=66−16=56P(\text{not six}) = 1 - \frac{1}{6} = \frac{6}{6} - \frac{1}{6} = \frac{5}{6}

The correct option is (b).

(b) 5/6\boxed{\text{(b) 5/6}}

Part ix

P(A)+P(Aˉ)P(A) + P(\bar{A}) is equal to

P(A)P(A) is the probability of an event A occurring, and P(Aˉ)P(\bar{A}) (or P(A′)P(A')) is the probability of event A not occurring (its complement). The sum of the probability of an event and the probability of its complement is always 1.
The correct option is (c).

(c) 1\boxed{\text{(c) 1}}

Part x· 2 steps

A fair die and a fair coin are thrown at a time. The number of combined outcomes is

  1. Determine the number of outcomes for each independent event.

    Number of outcomes for a fair die = 6 (i.e., {1,2,3,4,5,6}\{1, 2, 3, 4, 5, 6\}).
    Number of outcomes for a fair coin = 2 (i.e., {Head,Tail}\{Head, Tail\}).

  2. Calculate the total number of combined outcomes.

    For independent events, the total number of combined outcomes is the product of the number of outcomes for each event.
    Total combined outcomes = (Outcomes for die) ×\times (Outcomes for coin) = 6×2=126 \times 2 = 12.
    The correct option is (b).

    (b) 12\boxed{\text{(b) 12}}
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