What is the probability that it will be on time all five times

[lovemake1]

Homework Statement

1. Joshua's afternoon bus is on-time 60% of the time. What is the
probability that it will be on time all five times this week?
State any assumptions that you are relying on.2. In this trial, a fair 6-sided die will be rolled until two consecutive sixes
have been observed. The number of rolls (r) will be recorded.
What is the sample space for this trial?

Homework Equations

The Attempt at a Solution

1. since there is 0.6 cahnce bus will be on time, is the answer 0.6^5?
I'm not sure what it means by "assumptions" like weather will be good all week?

2. Would the sample space be infinite? since it can technically never repeat 6 twice. thanks in advance

 

[Ibix]

1 - 0.6⁵ is as good an answer as you can get from the information available. As to the assumptions, let me pose a similar problem. I work three days per week - so if you pick a weekday at random, there is an 60% chance that I am in the office. What is the probability that I am in the office all five days this week?

2 - The sample space is the set of all possible outcomes - so "it's infinite" isn't a possible answer Edit: although the set can have an infinite number of members, which is what you were asking (thanks, rcgldr). An outcome in this case is the number, r, of rolls of the die needed to get two consecutive sixes. Obviously r is a non-negative integer, as you can't have negative or fractional rolls. What other limits can you put on it?

 

[rcgldr]

2 - The sample space is the set of all possible outcomes

This would depend on r, the number of rolls. I think what the OP is getting at is that it's possible but with very small probablity that two 6's in a row never occurs in any finite number of rolls.

Is it allowed to specify the sample space as a function of r?

 

[Ray Vickson]

Homework Statement

1. Joshua's afternoon bus is on-time 60% of the time. What is the
probability that it will be on time all five times this week?
State any assumptions that you are relying on.


2. In this trial, a fair 6-sided die will be rolled until two consecutive sixes
have been observed. The number of rolls (r) will be recorded.
What is the sample space for this trial?

Homework Equations

The Attempt at a Solution

1. since there is 0.6 cahnce bus will be on time, is the answer 0.6^5?
I'm not sure what it means by "assumptions" like weather will be good all week?

2. Would the sample space be infinite? since it can technically never repeat 6 twice.


thanks in advance

For question 2:

Of course it can repeat 6 twice, or three times, or 100 times, etc.; there is no law of Physics that says you cannot get '66' or '666' or '6666666666666' during successive tosses of a die. However, the more 6s you want in a row, the longer you will have to wait to see that pattern. For example, if T2 = number of tosses until you see '66' for the first time, the mean is E(T2) = 42 (that is, you must perform 42 tosses on average). However, the probability distribution of T2 is skewed, so the mean does tell you much. The probability of seeing '66' on or before toss 42 is only 0.63665, so there is about a 36% chance it will not happen yet by toss 42. If T4 = number of tosses until you see '6666' for the first time, then the mean is E(T4) = 1554 (so on average, you need about 1500 tosses to see the pattern for the first time), and working out actual probabilities is challenging.

You were correct in saying that the sample space must be infinite, since in principle there is always a nonzero chance of not having two 6s in a row within the first million, or billiion, or ... tosses. However, the probability of not seeing the pattern during the first N tosses goes to zero as N goes to infinity. In other words, the probability of seeing the pattern at least once in N tosses goes to 1 as N goes to infinity. Furthermore, the probability of seeing it infinitely many times in infinitely many tosses is 1!

RGV

 

[HallsofIvy]

Homework Statement

1. Joshua's afternoon bus is on-time 60% of the time. What is the
probability that it will be on time all five times this week?
State any assumptions that you are relying on.

.6⁶ is correct assuming that the arrival of the bus from day to day is independent of what happened on previous days.

2. In this trial, a fair 6-sided die will be rolled until two consecutive sixes
have been observed. The number of rolls (r) will be recorded.
What is the sample space for this trial?

Homework Equations

The Attempt at a Solution

1. since there is 0.6 cahnce bus will be on time, is the answer 0.6^5?
I'm not sure what it means by "assumptions" like weather will be good all week?

2. Would the sample space be infinite? since it can technically never repeat 6 twice.


thanks in advance