Q.Define outcomes and sample space. The numbers 1, 2, 3 and 4 are written separately on four slips of paper. The slips are put in a box and mixed thoroughly. A person draws two slips from the box, one after the other, without replacement. Describe the sample space for the experiment. (1+1+4=6) OR Two dice are thrown and the sum of the numbers which come up on the dice is noted. Consider the following events associated with this experiment: A: 'the sum is even'; B: 'the sum is a multiple of 3'; C: 'the sum is less than 4'; D: 'the sum is greater than 11'. Which pairs of these events are mutually exclusive?

For $n$ independent tosses of a coin, the sample space has $2^n$ equally likely outcomes.

Each toss has 2 outcomes, Head (H) or Tail (T). For 3 tosses, the total number of outcomes is $2\times2\times2=2^3=8$.

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Two coins tossed together have $4$ sample points.

Each coin independently has $2$ outcomes (Head, Tail). By the multiplication principle, two coins together have $2\times2=4$ outcomes …

Two dice thrown together have $36$ sample points.

Each die has $6$ faces $(1,2,3,4,5,6)$. By the multiplication principle, two dice give …

By definition, an event with only one sample point in the sample space is called a simple or elementary event.

In probability theory, if an event $E$ associated with a random experiment contains only a single outcome (sample point) of the sample space, it is called a simple event (also called an elementary even …

By definition, an event containing more than one sample point of the sample space is called a compound event.

In probability, events are classified based on the number of sample points they contain:

• An event with exactly one sample point is a simple (elementary) event. …

The sample space for tossing a coin thrice has 8 outcomes: $S=\{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}$.

Each toss independently results in Head (H) or Tail (T), giving 2 possibilities per toss. For 3 tosses, the total number of outcomes is $2\times2\times2=8$. Listing all …

Tossing a coin twice gives the sample space $\{HH, HT, TH, TT\}$, 4 equally likely outcomes.

Each individual coin toss has 2 possible outcomes: H (Heads) or T (Tails). When the coin is tossed twice, the outcome is an ordered pair (first toss, second toss), so the total number of ou …

Each coin independently lands Head or Tail (2 outcomes); by the multiplication principle, $n$ coins give $2\times2\times\cdots\times2$ ($n$ times).

Each of the $n$ coins has 2 possible outcomes (Head or Tail), and the coins are tossed independently.

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The sample space has $2^3=8$ equally likely outcomes.

Each toss has 2 possible outcomes (H or T), and there are 3 tosses, so by the multiplication principle there are $2^3=8$ outcomes: …

A simple event corresponds to a single outcome of the sample space.

For a random experiment with sample space $S$, an event is any subset of $S$. If an event $E$ contains exactly one element of $S$ (i.e., it corresponds to only one possible outcome of the experiment), it is called a simple event or elementary event.

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