I'm trying to calculate probablity of dependent and independent events. For example, from 3 nickels, 2 dimes, and 5 quarters, what is the probability of drawing 1 nickel, 1 dime, and 1 quarter in that order without replacement? Or drawing the same in any order with replacement? I don't understand the difference in how to calculate them. Thanks for any help!
Without replacement, the probability of drawing the nickel is 3/10, the dime is 2/9 and the quarter is 5/8. Mulitply the probability of the three events to get the probability that all would occur. Each probability is dependent on the previous event.
With replacement, the same probabilities are 3/10, 2/10 and 5/10. Multiply again. The probabilities are independent of each other, so order is not important.
I hope this helps. Thanks for asking.
Very helpful! Thank you!
As another part of the same type question...what if they have to be selected in a certain order (1 nickel, then 1 dime, then 1 quarter) as opposed to a nickel, a dime, and a quarter in any order? Thanks for your time!
To calculate the probability of dependent events (without replacement), you need to consider that the probability of the second event depends on the outcome of the first event, as the items are not replaced after each draw.
Let's start with the first scenario: drawing 1 nickel, 1 dime, and 1 quarter in that order without replacement.
Step 1: Calculate the probability of drawing a nickel first.
There are 3 nickels out of a total of 10 coins, so the probability of drawing a nickel first is 3/10.
Step 2: Calculate the probability of drawing a dime second, after drawing a nickel.
After drawing a nickel, there are 2 dimes left out of a total of 9 remaining coins. Therefore, the probability of drawing a dime second is 2/9.
Step 3: Calculate the probability of drawing a quarter third, after drawing a nickel and a dime.
After drawing a nickel and a dime, there are 5 quarters left out of a total of 8 remaining coins. So the probability of drawing a quarter third is 5/8.
Step 4: Multiply the probabilities from each step.
To find the overall probability of drawing 1 nickel, 1 dime, and 1 quarter in that order without replacement, we multiply the probabilities from each step together: (3/10) * (2/9) * (5/8) = 1/12.
Now let's move on to the second scenario: drawing the same in any order with replacement.
Step 1: Calculate the probability of drawing a nickel.
There are 3 nickels out of a total of 10 coins, so the probability of drawing a nickel is 3/10.
Step 2: Calculate the probability of drawing a dime.
There are 2 dimes out of a total of 10 coins, so the probability of drawing a dime is 2/10 (or 1/5).
Step 3: Calculate the probability of drawing a quarter.
There are 5 quarters out of a total of 10 coins, so the probability of drawing a quarter is 5/10 (or 1/2).
Step 4: Multiply the probabilities from each step.
Since the events are independent and with replacement, the probabilities from each step can be multiplied together: (3/10) * (1/5) * (1/2) = 3/100.
Therefore, the probability of drawing 1 nickel, 1 dime, and 1 quarter in any order with replacement is 3/100.
To summarize:
1. Probability of drawing 1 nickel, 1 dime, and 1 quarter in that order without replacement: 1/12.
2. Probability of drawing 1 nickel, 1 dime, and 1 quarter in any order with replacement: 3/100.
To calculate the probability of dependent and independent events, we need to understand the concepts of dependent and independent events.
Dependent events are events where the outcome of one event affects the outcome of another event. In your first scenario, drawing 1 nickel, 1 dime, and 1 quarter without replacement, the events are dependent because the probability of drawing each coin changes after each draw. Once a coin is drawn, the total number of coins decreases, affecting the probability of the next draw.
Independent events are events where the outcome of one event does not affect the outcome of another event. In your second scenario, drawing the same coins in any order with replacement, the events are independent because each draw of a coin does not affect the probability of the next draw. After each draw, the coin is returned to the set of coins, and the probability remains the same for each subsequent draw.
Let's calculate the probabilities for both scenarios:
Scenario 1: Drawing 1 nickel, 1 dime, and 1 quarter without replacement
To calculate the probability for each step, we need to consider the number of coins remaining for each draw.
P(drawing a nickel) = (Number of nickels / Total number of coins) = (3/10) = 0.3
P(drawing a dime after drawing a nickel) = (Number of dimes / Remaining number of coins) = (2/9) = 0.222
P(drawing a quarter after drawing a nickel and a dime) = (Number of quarters / Remaining number of coins) = (5/8) = 0.625
To find the probability of all three events happening in that order without replacement, we multiply the individual probabilities:
P(drawing 1 nickel, 1 dime, and 1 quarter without replacement) = P(drawing a nickel) * P(drawing a dime after drawing a nickel) * P(drawing a quarter after drawing a nickel and a dime)
= 0.3 * 0.222 * 0.625
= 0.04185 (approximately)
So the probability of drawing 1 nickel, 1 dime, and 1 quarter in that order without replacement is approximately 0.04185.
Scenario 2: Drawing the same coins in any order with replacement
In this scenario, each draw is independent, and the probability remains the same after each draw.
P(drawing a nickel) = (Number of nickels / Total number of coins) = (3/10) = 0.3
P(drawing a dime) = (Number of dimes / Total number of coins) = (2/10) = 0.2
P(drawing a quarter) = (Number of quarters / Total number of coins) = (5/10) = 0.5
To find the probability of drawing the same coins in any order with replacement, we need to consider all possible orders of drawing the coins. In this case, there are 3! (3 factorial) possible orders, as there are 3 distinct coins.
P(drawing the same coins in any order with replacement) = P(drawing 1 nickel, 1 dime, and 1 quarter) + P(drawing 1 nickel, 1 quarter, and 1 dime) + P(drawing 1 dime, 1 nickel, and 1 quarter) + P(drawing 1 dime, 1 quarter, and 1 nickel) + P(drawing 1 quarter, 1 nickel, and 1 dime) + P(drawing 1 quarter, 1 dime, and 1 nickel)
= (0.3 * 0.2 * 0.5) + (0.3 * 0.5 * 0.2) + (0.2 * 0.3 * 0.5) + (0.2 * 0.5 * 0.3) + (0.5 * 0.3 * 0.2) + (0.5 * 0.2 * 0.3)
= 0.03 + 0.03 + 0.03 + 0.03 + 0.03 + 0.03
= 0.18
So the probability of drawing the same coins in any order with replacement is 0.18.
I hope this explanation helps! Let me know if you have any further questions.