1. Dependent
2. Independent
3. Mutually exclusive
4. Mutually inclusive
Independent Events: Definition and Explanation
This question asks for the term used when two events are not influenced by each other's outcomes. Let's explore the definitions of the given options:
Understanding Event Relationships
- Independent Events: Two events are considered independent if the occurrence or non-occurrence of one event does not affect the probability of the other event occurring. For example, flipping a coin twice. The result of the first flip (Heads or Tails) has absolutely no impact on the result of the second flip.
- Dependent Events: Conversely, two events are dependent if the outcome of one event does change the probability of the other event. For instance, drawing two cards from a deck without replacement. The outcome of the first draw affects the probabilities for the second draw because the first card is not put back in the deck.
- Mutually Exclusive Events: These are events that cannot happen at the same time. For example, when flipping a coin once, the outcomes Heads and Tails are mutually exclusive; you cannot get both simultaneously.
- Mutually Inclusive Events: This term isn't standard in probability theory like the others. It might informally refer to events that *can* occur together, but the key concept in the question is about influence, not just simultaneous possibility.
Comparing Event Types
The defining characteristic mentioned in the question – that one event is unaffected by the outcome of another – precisely matches the definition of independent events.
| Event Type | Definition | Example |
|---|---|---|
| Independent | Outcome of one event does NOT affect the outcome of another. | Flipping a coin twice. |
| Dependent | Outcome of one event DOES affect the outcome of another. | Drawing cards without replacement. |
| Mutually Exclusive | Events cannot occur at the same time. | Getting Heads and Tails on a single coin flip. |
Therefore, when one event is unaffected by the outcome of another, the two events are called independent.
Important Questions from Probability
- Let $M$ be a randomly chosen non-empty subset of $S = \{1, 2, 3, \dots, 2026\}$.
Which of the following is the probability that the product of all the elements of $M$ is even? - A fair coin is tossed 6 times. The probability of getting at least 5 heads is _________________. (rounded off to two decimal places)
- Consider a random experiment of rolling a fair die. The probability of getting an odd number or a number less than 4 is ______ (Rounded off to two decimal places).
- In a null hypothesis significance test, the significance level $\alpha$ was set to 0.01. For which one or more of the following $p$ values would the null hypothesis be rejected?
In a population of birds, the proportion of blue individuals is 0.4 and the proportion of males is 0.3. Assuming that colour and sex are independent of each other, the probability that a randomly sampled individual from this population is a blue male is ______
(Round off to two decimal places)
