Introduction to Probability Calculation in a Simple Game
This article delves into the detailed process of calculating the probability of drawing one white and one red ball from a bag containing 3 white and 5 red balls. We will explore the step-by-step method and provide a comprehensive guide on how to calculate such probabilities.
Understanding the Basic Concept
Let's start with a basic understanding of what probability means in this context. Probability is the measure of the likelihood that an event will occur. In this case, we are interested in the probability of drawing one white and one red ball from a bag containing a total of 3 white balls and 5 red balls. This combination of events can occur in two possible orders: either a white ball first, followed by a red ball, or a red ball first, followed by a white ball.
Probability Calculation: Order-Specific Analysis
First, let's analyze the probability of drawing one white and one red ball in a specific order: a white ball first, followed by a red ball.
Step-by-Step Calculation for White then Red
1. Total number of balls: The bag initially contains a total of 8 balls (3 white and 5 red).
2. Probability of drawing the first white ball: The probability of drawing a white ball first is given by the number of white balls divided by the total number of balls.
[ P(white first) frac{3}{8} ]
3. Updated count of balls: After drawing one white ball, there are now 7 balls left in the bag (2 white and 5 red).
4. Probability of drawing the second red ball: The probability of drawing a red ball second is now the number of red balls divided by the remaining total number of balls.
[ P(red second) frac{5}{7} ]
5. Combined probability: The combined probability of both events occurring in this order is the product of the two probabilities calculated above.
[ P(white then red) frac{3}{8} times frac{5}{7} frac{15}{56} ]
Now, let's consider the reverse order: a red ball first, followed by a white ball.
Step-by-Step Calculation for Red then White
1. Total number of balls: The bag initially contains a total of 8 balls (3 white and 5 red).
2. Probability of drawing the first red ball: The probability of drawing a red ball first is given by the number of red balls divided by the total number of balls.
[ P(red first) frac{5}{8} ]
3. Updated count of balls: After drawing one red ball, there are now 7 balls left in the bag (3 white and 4 red).
4. Probability of drawing the second white ball: The probability of drawing a white ball second is now the number of white balls divided by the remaining total number of balls.
[ P(white second) frac{3}{7} ]
5. Combined probability: The combined probability of both events occurring in this order is the product of the two probabilities calculated above.
[ P(red then white) frac{5}{8} times frac{3}{7} frac{15}{56} ]
Combining Both Orders
Since the order in which we draw the balls can be either white first or red first, we need to add the probabilities of both events to find the total probability of drawing one white and one red ball.
[ P(one white and one red) P(white then red) P(red then white) frac{15}{56} frac{15}{56} frac{30}{56} frac{15}{28} ]
Probability of Drawing Both Balls Red
Lastly, let's calculate the probability of drawing two red balls, assuming the first one is not replaced. This is a scenario where both draws are independent events.
Step-by-Step Calculation for Two Red Balls
1. Total number of balls: The bag initially contains a total of 8 balls (3 white and 5 red).
2. Probability of drawing the first red ball: The probability of drawing a red ball first is given by the number of red balls divided by the total number of balls.
[ P(first red) frac{5}{8} ]
3. Updated count of balls: After drawing one red ball, there are now 7 balls left in the bag (2 white and 4 red).
4. Probability of drawing the second red ball: The probability of drawing a red ball second is now the number of remaining red balls divided by the remaining total number of balls.
[ P(second red) frac{4}{7} ]
5. Combined probability: The combined probability of both events occurring in this order is the product of the two probabilities calculated above.
[ P(two red) frac{5}{8} times frac{4}{7} frac{20}{56} frac{10}{28} frac{5}{14} ]
The final answer for the probability of drawing one white and one red ball is $boxed{frac{15}{28}}$ , and the probability of drawing both balls red is $boxed{frac{5}{14}}$.
In summary, the article provides a detailed and methodical approach to calculate the probability of drawing one white and one red ball, and both balls red, from a bag containing a specific number of white and red balls. This step-by-step analysis ensures a clear understanding of the underlying concepts and can be applied to similar probability problems.