Problem 9 - Algebra 2

Question

Determine whether the events are independent or dependent. Then find the probability. Yana has 4 black socks, 6 blue socks, and 8 white socks in his drawer. If he selects three socks at random with no replacement, what is the probability that he will first select a blue sock, then a black sock, and then another blue sock?

Step-by-Step Solution

Answer

The events are dependent, and the probability of the sequence is \( \frac{5}{204} \).

Step 1: Determine if Events are Independent or Dependent

Two events are independent if the occurrence of one does not affect the probability of the occurrence of the other. Since Yana is drawing without replacement, each selection affects the subsequent draw. Thus, these are dependent events.

Step 2: Calculate Total Number of Socks

First, calculate the total number of socks, which is the sum of the black, blue, and white socks: \[ 4 + 6 + 8 = 18 \] socks in total.

Step 3: Calculate Probability of Selecting the First Blue Sock

There are 6 blue socks out of 18 total socks. The probability of drawing a blue sock first is:\[ P(\text{Blue}_1) = \frac{6}{18} = \frac{1}{3} \]

Step 4: Calculate Probability of Selecting the Second Black Sock

After selecting a blue sock, there are now 17 socks remaining, of which 4 are black. The probability of drawing a black sock second is:\[ P(\text{Black}_2 | \text{Blue}_1) = \frac{4}{17} \]

Step 5: Calculate Probability of Selecting the Second Blue Sock

After selecting a blue and a black sock, there are now 16 socks remaining, with 5 of them being blue. The probability of drawing a second blue sock third is:\[ P(\text{Blue}_3 | \text{Black}_2 \text{ and } \text{Blue}_1) = \frac{5}{16} \]

Step 6: Calculate Overall Probability

To find the overall probability of drawing in the sequence blue-black-blue, multiply the probabilities from each step:\[ P(\text{Blue}_1, \text{Black}_2, \text{Blue}_3) = \frac{1}{3} \times \frac{4}{17} \times \frac{5}{16} = \frac{20}{816} = \frac{5}{204} \]

Key Concepts

Dependent Events

In probability, understanding whether events are dependent or independent is crucial in calculating the overall likelihood of combined outcomes. Events are dependent when the outcome of one event affects the probability of another event happening.
This means that one event changes the circumstance or alters the possible outcomes of the following event. In the original problem, Yana draws socks from a drawer without replacement.
- The absence of replacement indicates that the total number of socks changes with each draw. - This affects the probabilities of subsequent draws because each action (drawing a sock) directly influences the situation (fewer socks remain). For instance, when Yana draws a blue sock first, there are fewer total socks for the next pick. - More importantly, the number of blue socks also reduces, affecting the odds of picking another blue sock later.
Understanding this dependence aids in accurately determining the likelihood of a sequence of events.

Independent Events

Independent events are those in which the result or happening of one event doesn't impact another. Each event has no effect on the probabilities of the subsequent events occurring. A classic example would be flipping a coin: each toss of a coin doesn’t affect the chances of heads or tails on the next toss.
In scenarios involving picks, draws, or choices where the items are *with replacement*, events tend to be independent.
- When an item is replaced, the conditions for each pick return to their original state. - Thus, making each event entirely unrelated in terms of probability distribution. Returning socks into the drawer after each draw would have made Yana’s sock selections independent. However, in our original exercise, Yana did not replace the socks, so they are dependent.

Probability Without Replacement

Probability without replacement is a method used to calculate the likelihood of a sequence of events where items are not returned to the pool of possibilities for subsequent selections. This is common in real-life scenarios where chosen items are not put back or returned before the next selection.
When you draw without replacement, each item removed affects the total count and composition of the remaining selection. This is why we see changing denominators in each probability stage of Yana's sock selection problem. Here's a quick recap of how we manage these calculations: - **First Event:** Identify the probability based on the initial condition (total items). - **Subsequent Events:** Adjust the remaining total after each selection impacts the pool.
Each step builds on the previous one, altering probabilities, which makes this process unique from calculations involving replacement. For example, after Yana's first draw of a blue sock, only 17 socks remained, impacting the subsequent draw calculation.