Respuesta :
Answer:
(a) [tex]\frac{1}{216}[/tex]
(b) [tex]\frac{1}{8}[/tex]
(c) [tex]\frac{1}{8}[/tex]
(d) [tex]\frac{191}{216}[/tex]
(e) [tex]\frac{215}{216}[/tex]
Step-by-step explanation:
It is given that we roll a fair die three times. We need to find the following probability.
Total possible value = 1,2,3,4,5,6
(a) You roll all 4's.
Probability of getting a 4 = [tex]\frac{1}{6}[/tex]
Probability of getting 4 on all rolls = [tex]\frac{1}{6}\times \frac{1}{6}\times \frac{1}{6}=\frac{1}{216}[/tex]
Therefore, the probability of getting 4 on all rolls is [tex]\frac{1}{216}[/tex].
(b) You roll all even numbers.
Even number = 2,4,6
Probability of getting an even number [tex]=\frac{3}{6}=\frac{1}{2}[/tex]
Probability of getting even number on all rolls = [tex]\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}=\frac{1}{8}[/tex]
Therefore, the probability of getting even number on all rolls is [tex]\frac{1}{8}[/tex].
(c) None of your rolls gets a number divisible by 2.
odd number = 1,3,5
Probability of getting an odd number [tex]=\frac{3}{6}=\frac{1}{2}[/tex]
Probability of getting odd number on all rolls = [tex]\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}=\frac{1}{8}[/tex]
Therefore, the probability of getting odd number on all rolls is [tex]\frac{1}{8}[/tex].
(d) You roll at least one 2.
Probability of getting a 2 = [tex]\frac{1}{6}[/tex]
Probability of getting any number except 2 = [tex]\frac{5}{6}[/tex]
Probability of getting any number except 2 in all rolls = [tex]\frac{5}{6}\times \frac{5}{6}\times \frac{5}{6}=\frac{125}{216}[/tex]
Probability of getting at least one 2 = 1 - P(getting no 2)
Probability of getting at least one 2 = [tex]1-\frac{125}{216}=\frac{191}{216}[/tex]
Therefore, the probability of getting at least one 2 is [tex]\frac{191}{216}[/tex].
(e) The numbers you roll are not all 2's.
Probability of getting a 2 = [tex]\frac{1}{6}[/tex]
Probability of getting all 2 = [tex]\frac{1}{6}\times \frac{1}{6}\times \frac{1}{6}=\frac{1}{216}[/tex]
Probability of getting all not all 2's = [tex]1-\frac{1}{216}=\frac{215}{216}[/tex]
Therefore, the probability of getting not all 2's is [tex]\frac{215}{216}[/tex].
