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)
Probability Calculation for Independent Events
The problem involves calculating the probability of two independent events occurring simultaneously. We are given the individual probabilities and told the events are independent.
Given Probabilities
- The probability of a bird being blue, $P(\text{Blue})$, is 0.4.
- The probability of a bird being male, $P(\text{Male})$, is 0.3.
Calculating Joint Probability
Since colour (blue) and sex (male) are stated to be independent, the probability of both events occurring (a bird being a blue male) is the product of their individual probabilities.
The formula for independent events A and B is: $P(A \text{ and } B) = P(A) \times P(B)$
Applying this to our problem:
$P(\text{Blue and Male}) = P(\text{Blue}) \times P(\text{Male})$ $P(\text{Blue and Male}) = 0.4 \times 0.3$ $P(\text{Blue and Male}) = 0.12$Final Result
The calculated probability is 0.12. Rounding this to two decimal places results in 0.12.
Important Questions from Probability
- If one event is unaffected by the outcome of another event, the two events are said to be:
1. Dependent
2. Independent
3. Mutually exclusive
4. Mutually inclusive - 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?
