Welcome to the article A box contains 5 red pens and 3 blue pens Two pens are randomly picked from the box one at a time without replacement a. On this page, you will learn the essential and logical steps to better understand the topic being discussed. We hope the information provided helps you gain valuable insights and is easy to follow. Let’s begin the discussion!
Answer :
To solve this problem and illustrate it using a tree diagram with probabilities, we need to consider the step-by-step process of selecting pens from the box without replacement. Here's how we can approach this:
1. Initial probabilities for the first pick:
- Total number of pens = 5 red + 3 blue = 8 pens.
- Probability of picking a red pen first:
[tex]\[
\frac{\text{Number of red pens}}{\text{Total pens}} = \frac{5}{8} = 0.625
\][/tex]
- Probability of picking a blue pen first:
[tex]\[
\frac{\text{Number of blue pens}}{\text{Total pens}} = \frac{3}{8} = 0.375
\][/tex]
2. Probabilities for the second pick depending on the first pick (without replacement):
- If the first pen picked is red:
- Remaining pens: 4 red, 3 blue (total 7 pens).
- Probability of picking another red pen:
[tex]\[
\frac{\text{Remaining red pens}}{\text{Total remaining pens}} = \frac{4}{7} \approx 0.571
\][/tex]
- Probability of picking a blue pen:
[tex]\[
\frac{\text{Remaining blue pens}}{\text{Total remaining pens}} = \frac{3}{7} \approx 0.429
\][/tex]
- If the first pen picked is blue:
- Remaining pens: 5 red, 2 blue (total 7 pens).
- Probability of picking a red pen:
[tex]\[
\frac{\text{Remaining red pens}}{\text{Total remaining pens}} = \frac{5}{7} \approx 0.714
\][/tex]
- Probability of picking another blue pen:
[tex]\[
\frac{\text{Remaining blue pens}}{\text{Total remaining pens}} = \frac{2}{7} \approx 0.286
\][/tex]
3. Tree Diagram:
The tree diagram will have two levels:
- First Level: Represents the first pick:
- Red (probability = 0.625)
- Blue (probability = 0.375)
- Second Level: Represents the second pick, branching from each outcome of the first pick:
- From Red:
- Red (probability = 0.571)
- Blue (probability = 0.429)
- From Blue:
- Red (probability = 0.714)
- Blue (probability = 0.286)
These probabilities can be depicted in a tree diagram showing the sequence of events and their probabilities. This structured way helps in visualizing and understanding the different outcomes when selecting pens from the box without replacement.
1. Initial probabilities for the first pick:
- Total number of pens = 5 red + 3 blue = 8 pens.
- Probability of picking a red pen first:
[tex]\[
\frac{\text{Number of red pens}}{\text{Total pens}} = \frac{5}{8} = 0.625
\][/tex]
- Probability of picking a blue pen first:
[tex]\[
\frac{\text{Number of blue pens}}{\text{Total pens}} = \frac{3}{8} = 0.375
\][/tex]
2. Probabilities for the second pick depending on the first pick (without replacement):
- If the first pen picked is red:
- Remaining pens: 4 red, 3 blue (total 7 pens).
- Probability of picking another red pen:
[tex]\[
\frac{\text{Remaining red pens}}{\text{Total remaining pens}} = \frac{4}{7} \approx 0.571
\][/tex]
- Probability of picking a blue pen:
[tex]\[
\frac{\text{Remaining blue pens}}{\text{Total remaining pens}} = \frac{3}{7} \approx 0.429
\][/tex]
- If the first pen picked is blue:
- Remaining pens: 5 red, 2 blue (total 7 pens).
- Probability of picking a red pen:
[tex]\[
\frac{\text{Remaining red pens}}{\text{Total remaining pens}} = \frac{5}{7} \approx 0.714
\][/tex]
- Probability of picking another blue pen:
[tex]\[
\frac{\text{Remaining blue pens}}{\text{Total remaining pens}} = \frac{2}{7} \approx 0.286
\][/tex]
3. Tree Diagram:
The tree diagram will have two levels:
- First Level: Represents the first pick:
- Red (probability = 0.625)
- Blue (probability = 0.375)
- Second Level: Represents the second pick, branching from each outcome of the first pick:
- From Red:
- Red (probability = 0.571)
- Blue (probability = 0.429)
- From Blue:
- Red (probability = 0.714)
- Blue (probability = 0.286)
These probabilities can be depicted in a tree diagram showing the sequence of events and their probabilities. This structured way helps in visualizing and understanding the different outcomes when selecting pens from the box without replacement.
Thank you for reading the article A box contains 5 red pens and 3 blue pens Two pens are randomly picked from the box one at a time without replacement a. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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Rewritten by : Brahmana