Probability of Six on Next Throw
The core principle to understand here is the concept of independent events in probability.
- A fair die has 6 faces, numbered 1 through 6.
- Each face has an equal probability of landing face up on any given throw.
- Crucially, each throw of the die is an independent event. This means the outcome of previous throws has absolutely no influence on the outcome of the next throw.
The fact that the die landed on six three times in a row does not change the inherent probabilities for the *next* throw.
Calculating Single Event Probability
For a fair die, the probability of rolling any specific number (like a six) is calculated as:
$P(\text{Specific Outcome}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$In this case, for rolling a six:
$P(\text{rolling a 6}) = \frac{1}{6}$Conclusion for Next Throw
Since the next throw is independent of the previous ones, the probability of getting a six remains the same as it was for the first throw.
Therefore, the probability of getting a six on the next throw is $1/6$.
Important Questions from Probability (Notes)
- In a box there are 4 white balls and 6 black balls. A ball is drawn at random. If it is white, it is put back along with two more white balls in the box. If it is black, it is put back in the box and then two black balls are thrown out of the box. Now a ball is drawn again at random from the box. Then, what is the probability that it is black?
- A fair coin is tossed three times. Let A be the event of getting exactly two heads and B be the event of getting at most
two tails, then P(A$\cup$B) is: Bag A contains 3 Red and 4 Black balls while Bag B contains 5 Red and 6 Black balls. One ball is drawn at random from one of the bags and is found to be red. Then, the probability that it was drawn from Bag B is
- If we twice flip a balanced coin, what is the probability of getting at least one head?
1. 1/4
2. 2/4
3. 1/6
4. 3/4 Suppose that the random variable X takes on the values: -1, 0, and 2 with probability $\frac{1}{8}$, $\frac{1}{2}$ and $\frac{3}{8}$. Find the expected value of X.
