A teacher has 6 red balls, 7 blue balls, 8 purple balls, and 4 black balls in a basket. A student reaches into the basket and randomly selects a ball. What is the probability that the ball will be either blue or black?
Probability Calculation for Ball Selection
Understanding the concept of probability is crucial for solving such problems. Probability measures the likelihood of an event occurring. It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. In this scenario, a student is selecting a ball from a basket, and we need to find the probability of selecting a blue or a black ball.
Balls in the Basket: Determining Total Outcomes
First, let's determine the total number of balls present in the basket. The basket contains different colors of balls, and we need to sum them up to find the total possible outcomes when a student selects a ball.
- Number of red balls: 6
- Number of blue balls: 7
- Number of purple balls: 8
- Number of black balls: 4
The total number of balls in the basket is the sum of balls of all colors:
Total balls = (Number of red balls) + (Number of blue balls) + (Number of purple balls) + (Number of black balls)
Total balls = \(6 + 7 + 8 + 4 = 25\)
So, there are 25 total possible outcomes when a ball is randomly selected.
Blue and Black Balls: Identifying Favorable Outcomes
Next, we need to identify the number of favorable outcomes. The question asks for the probability that the ball will be either blue or black. This means any blue ball or any black ball selected would be considered a favorable outcome.
- Number of blue balls: 7
- Number of black balls: 4
The number of favorable outcomes (selecting a blue or black ball) is the sum of the number of blue balls and the number of black balls:
Favorable outcomes = (Number of blue balls) + (Number of black balls)
Favorable outcomes = \(7 + 4 = 11\)
So, there are 11 favorable outcomes.
Calculating Probability: Blue or Black Ball
Now, we can calculate the probability using the probability formula:
Probability = \(\frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}\)
Substituting the values we found:
Probability (blue or black) = \(\frac{11}{25}\)
To summarize the calculation:
| Category | Quantity |
|---|---|
| Red Balls | 6 |
| Blue Balls | 7 |
| Purple Balls | 8 |
| Black Balls | 4 |
| Total Balls | 25 |
| Favorable (Blue or Black) | 11 |
Therefore, the probability of selecting a ball that is either blue or black is \(\frac{11}{25}\).
Important Questions from Conditional Probability
A box contains 5 coins: 4 regular coins and 1 fake coin. When a regular coin is tossed, the probability 𝑃(ℎ𝑒𝑎𝑑) = 0.5 and for a fake coin, 𝑃(ℎ𝑒𝑎𝑑) = 1. You pick a coin at random and toss it twice, and get two heads. The probability that the coin you have chosen is the fake coin is _______. (rounded off to two decimal places)
- A clinic specializes in testing for a disease D. The result of the test can be either positive or negative.
A study revealed that if a person suffers from the disease D, the test result in that clinic comes out positive 80% of the time, and negative 20% of the time. If a person is not suffering from the disease D, the test comes out positive 10% of the time and negative 90% of the time. It is also known that among the general population, the disease D occurs in 30% of the individuals.
If a person tests positive for D in that clinic, the probability that he/she actually suffers from the disease D is __________ . (Rounded off to two decimal places) - Suppose an unbiased coin is tossed 6 times. Each coin toss is independent of all previous coin tosses. Let $E_1$ be the event that among the second, fourth, and sixth coin tosses, there are at least two heads. Let $E_2$ be the event that among the first, second, third, and fifth coin tosses, there are equal number of heads and tails.
The conditional probability P($E_1$ | $E_2$) is equal to ____________. (rounded off to one decimal place) - A box contains three apples and two oranges. Two fruits are removed randomly in succession. The probability that the first is an apple and the second is an orange is _____ .
- An experiment is performed by rolling an unbiased die once. Events A and B are defined as follows:
Event A: The outcome of the rolled die is a prime number.
Event B: The outcome of the rolled die is an odd number less than 4.
The conditional probability of A given B, denoted by P(A|B), is ______ (rounded off to one decimal place).
