Using the binomial distribution, it is found that there is a 0.857 = 85.7% probability that at least 2 of the rolls come up as a 3 or a 4.
For each die, there are only two possible outcomes, either a 3 or a 4 is rolled, or it is not. The result of a roll is independent of any other roll, hence, the <em>binomial distribution</em> is used to solve this question.
Binomial probability distribution
The parameters are:
- x is the number of successes.
- n is the number of trials.
- p is the probability of a success on a single trial.
In this problem:
- There are 9 rolls, hence
.
- Of the six sides, 2 are 3 or 4, hence
![p = \frac{2}{6} = 0.3333](https://tex.z-dn.net/?f=p%20%3D%20%5Cfrac%7B2%7D%7B6%7D%20%3D%200.3333)
The desired probability is:
![P(X \geq 2) = 1 - P(X < 2)](https://tex.z-dn.net/?f=P%28X%20%5Cgeq%202%29%20%3D%201%20-%20P%28X%20%3C%202%29)
In which:
![P(X < 2) = P(X = 0) + P(X = 1)](https://tex.z-dn.net/?f=P%28X%20%3C%202%29%20%3D%20P%28X%20%3D%200%29%20%2B%20P%28X%20%3D%201%29)
Then
![P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
![P(X = 0) = C_{9,0}.(0.3333)^{0}.(0.6667)^{9} = 0.026](https://tex.z-dn.net/?f=P%28X%20%3D%200%29%20%3D%20C_%7B9%2C0%7D.%280.3333%29%5E%7B0%7D.%280.6667%29%5E%7B9%7D%20%3D%200.026)
![P(X = 1) = C_{9,1}.(0.3333)^{1}.(0.6667)^{8} = 0.117](https://tex.z-dn.net/?f=P%28X%20%3D%201%29%20%3D%20C_%7B9%2C1%7D.%280.3333%29%5E%7B1%7D.%280.6667%29%5E%7B8%7D%20%3D%200.117)
Then:
![P(X < 2) = P(X = 0) + P(X = 1) = 0.026 + 0.117 = 0.143](https://tex.z-dn.net/?f=P%28X%20%3C%202%29%20%3D%20P%28X%20%3D%200%29%20%2B%20P%28X%20%3D%201%29%20%3D%200.026%20%2B%200.117%20%3D%200.143)
![P(X \geq 2) = 1 - P(X < 2) = 1 - 0.143 = 0.857](https://tex.z-dn.net/?f=P%28X%20%5Cgeq%202%29%20%3D%201%20-%20P%28X%20%3C%202%29%20%3D%201%20-%200.143%20%3D%200.857)
0.857 = 85.7% probability that at least 2 of the rolls come up as a 3 or a 4.
For more on the binomial distribution, you can check brainly.com/question/24863377
Answer:
13 centimeters per day.
Step-by-step explanation:
We have been given that during summer, the water level of the river increased by 78 centimeters in 6 days. We are asked to find the water level increase rate in centimeters per day.
To find the water level increase rate per day, we will divide total increase by total days as:
![\text{Water level increase rate per day}=\frac{78\text{ cm}}{6\text{ days}}](https://tex.z-dn.net/?f=%5Ctext%7BWater%20level%20increase%20rate%20per%20day%7D%3D%5Cfrac%7B78%5Ctext%7B%20cm%7D%7D%7B6%5Ctext%7B%20days%7D%7D)
![\text{Water level increase rate per day}=\frac{13\text{ cm}}{\text{ day}}](https://tex.z-dn.net/?f=%5Ctext%7BWater%20level%20increase%20rate%20per%20day%7D%3D%5Cfrac%7B13%5Ctext%7B%20cm%7D%7D%7B%5Ctext%7B%20day%7D%7D)
Therefore, the water level increase rate is 13 centimeters per day.
Answer:
700 meters
Step-by-step explanation: