Find alpha for this dice rolling game. There is the point A(10) on the number line. Let's following the die rolling game rules like below with its pips are 1 to 6 as commonly we've known.

Emmy Swanson 2022-10-23 Answered
Find α for this dice rolling game.
There is the point A(10) on the number line. Let's following the die rolling game rules like below with its pips are 1 to 6 as commonly we've known.
- The 1st rule) A point moves as much as +2[positive direction with magnitude 2] for the even pips like 2,4 and 6
- The 2nd rule) A point moves as much as −1[negative direction with magnitude 1] for the odd pips like 1,3 and 5.
After doing this trial 100 times, "A" lied on the right direction compared with 90. We get P ( Z α ) considering the continuity correction. Find the real number α. (Z is the random variable following normal distribution.)
From here, my solution starts. Let the X i be the movement of a point for ith trial (Here Each i, 1 i 100)
X i 2 1 p ( X i ) 1 / 2 1 / 2 1
From the table I got E ( X i ) = 1 2 and V ( X i ) = 9 4
So only have to consider is X ( = X 1 + X 2 + . . . + X 100 ) > 80, Plus E ( X ) = 50 and V ( X ) = 15 2
Therefore P ( X 80.5 ) = P ( Z 80.5 50 15 ) from continuity correction for P ( X > 80 ). (In other words, α = 61 30 ( = 30.5 15 ) )
The answer sheet claimed the α = 2.1. I can't find any errors in my solution. Please let me know which points did I wrong.
p.s.) The answer sheet solution. (The reason why the author claimed 2.1).Say Y be the number of the even pips doing the 100 times trials, Then only just find P ( 10 + 2 Y ( 100 Y ) > 90 )
In other words P ( Y 61 ). Since the Y~B(100,1/2), P ( Y 61 ) = P ( Y 60.5 ) = P ( z 2.1 )
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Answers (1)

Dana Simmons
Answered 2022-10-24 Author has 14 answers
Step 1
Note that P ( Y > 60 ) = P ( X > 90 ). The difference only comes into play when you add the continuity correction factor. The standard deviation of X is three times as great as the standard deviation of Y (because the distance between -1 and 2 is three times as great as the distance between 0 and 1). Therefore the continuity correction on X is 1 3 as significant as the continuity correction on Y.
Step 2
So I would say that both the "official" answer ( α = 2.1) and your solution ( α = 2.03333...) are reasonable approaches, with the difference being due to the relative importance of 1 unit of X compared to 1 unit of Y.
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