For a single roll of two dice, are rolling a sum of 6 and rolling doubles independent events? Explain.
Ten preschool children on a playground were asked to guess their mother’s age. Calculate the mean, median, and mode of their guesses.
39, 19, 34, 57, 100, 7, 44, 18, 25, 19
No, rolling a sum of 6 and rolling doubles are not independent events.
To understand why they are dependent, let's look at the different ways to roll a sum of 6 with two dice:
- Rolling a 1 and a 5 (1+5=6)
- Rolling a 2 and a 4 (2+4=6)
- Rolling a 3 and a 3 (3+3=6)
- Rolling a 4 and a 2 (4+2=6)
- Rolling a 5 and a 1 (5+1=6)
Out of these 5 possibilities, only one of them results in rolling doubles (3 and 3). So, if we know that the dice rolled a sum of 6, the probability of rolling doubles is 1 out of 5.
This demonstrates that the outcome of rolling a sum of 6 affects the probability of rolling doubles. Therefore, these events are dependent on each other.
To determine whether rolling a sum of 6 and rolling doubles are independent events when rolling two dice, we need to understand the concept of independence and how it applies to these two events.
In probability theory, two events are considered independent if the occurrence of one event does not affect the probability of the occurrence of the other event.
Let's break down the events:
1. Rolling a sum of 6: This event can occur in several ways when rolling two dice: (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1). So, there are a total of 5 favorable outcomes (out of 36 possible outcomes).
2. Rolling doubles: This event occurs when both dice show the same number. There are six possible outcomes: (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), and (6, 6).
Now, let's consider whether these events are independent:
To assess independence, we need to compare the probability of rolling a sum of 6 when doubles are already rolled with the original probability of rolling a sum of 6.
The probability of rolling a sum of 6 when doubles have already been rolled is 1 (since doubles means both dice have the same number). So, after rolling doubles, the event "rolling a sum of 6" will always happen with certainty.
However, the original probability of rolling a sum of 6 without considering doubles is 5/36.
Since the probability of rolling a sum of 6 changes after rolling doubles, it is clear that these events are not independent. The occurrence of rolling doubles affects the probability of rolling a sum of 6, making them dependent events.
In summary, rolling a sum of 6 and rolling doubles are not independent events when rolling two dice.
A -> Rolling a sum of 6 (Can occur with (1,5), (5,1), (3,3), (2,4), (4,2)
B -> Rolling Doubles (Can occur in 6 different ways)
P(A) = 5/36
P(B) = 6/36 = 1/6
P(A)*P(B) = (5/36)*(1/36)
= 5/1296
P(A∩B) = 1/36 (This only happens when (3,3) is rolled
In this case,
P(A)*P(B) =/= P(A∩B)
Hence, they are not independent, which means that their occurence affects the others' probability.