Three coins are tossed. Describe (i)Two events which are mutually exclusive. (ii)Three events which are mutually exclusive and exhaustive.(iii)Two events, which are not mutually exclusive.(iv)Two events which are mutually exclusive but not exhaustive.(v)Three events which are mutually exclusive but not exhaustive.
Three coins are tossed. Describe (i)Two events which are mutually exclusive. (ii)Three events which are mutually exclusive and exhaustive.(iii)Two events, which are not mutually exclusive.(iv)Two events which are mutually exclusive but not exhaustive.(v)Three events which are mutually exclusive but not exhaustive.
Solution and Explanation
When three coins are tossed, the sample space is given by
\(S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\} \)
(i) Two events that are mutually exclusive can be
A: getting no heads and B: getting no tails
This is because sets A \(= \{TTT\}\) and \(B = \{HHH\}\) are disjoint.
(ii) Three events that are mutually exclusive and exhaustive can be
A: getting no heads
B: getting exactly one head
C: getting at least two heads
i.e.,
\(A = \{TTT\} \)
\(B = \{HTT, THT, TTH\} \)
\(C = \{HHH, HHT, HTH, THH\} \)
This is because \(A ∩ B = B ∩ C = C ∩ A = Φ \text{ and } A ∪ B ∪ C = S\)
(iii) Two events that are not mutually exclusive can be
A: getting three heads
B: getting at least 2 heads
i.e.,
\(A = \{HHH\} \)
\(B = \{HHH, HHT, HTH, THH\} \)
This is because A ∩ B = \{HHH\} ≠Φ
(iv) Two events which are mutually exclusive but not exhaustive can be
A: getting exactly one head
B: getting exactly one tail That is
\(A = \{HTT, THT, TTH\} \)
\(B = \{HHT, HTH, THH\}\)
It is because, \(A ∩ B =Φ, \text{ but } A ∪ B ≠S\)
(v) Three events that are mutually exclusive but not exhaustive can be
A: getting exactly three heads
B: getting one head and two tails
C: getting one tail and two heads
i.e.,
\(A = \{HHH\} \)
\(B = \{HTT, THT, TTH\} \)
\(C = {HHT, HTH, THH} \)
This is because \(A ∩ B = B ∩ C = C ∩ A = Φ, \text{but }A ∪ B ∪ C ≠S\)
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Concepts Used:
Probability
Probability is defined as the extent to which an event is likely to happen. It is measured by the ratio of the favorable outcome to the total number of possible outcomes.
The definitions of some important terms related to probability are given below:
Sample space
The set of possible results or outcomes in a trial is referred to as the sample space. For instance, when we flip a coin, the possible outcomes are heads or tails. On the other hand, when we roll a single die, the possible outcomes are 1, 2, 3, 4, 5, 6.
Sample point
In a sample space, a sample point is one of the possible results. For instance, when using a deck of cards, as an outcome, a sample point would be the ace of spades or the queen of hearts.
Experiment
When the results of a series of actions are always uncertain, this is referred to as a trial or an experiment. For Instance, choosing a card from a deck, tossing a coin, or rolling a die, the results are uncertain.
Event
An event is a single outcome that happens as a result of a trial or experiment. For instance, getting a three on a die or an eight of clubs when selecting a card from a deck are happenings of certain events.
Outcome
A possible outcome of a trial or experiment is referred to as a result of an outcome. For instance, tossing a coin could result in heads or tails. Here the possible outcomes are heads or tails. While the possible outcomes of dice thrown are 1, 2, 3, 4, 5, or 6.