I threw a coin
n times, and never got three tails in a row. I calculated the odds of this event, and found out they were just about even; 50%-50%. How many times did I throw the coin?
A second question: what were the chances of having not gotten three heads in a row either?
The following program simulates 100,000 trials each at n = 3 through 12, and reports the results when, as in the stated condition, the tossing has continued until that many tosses have been made, under the assumption that as soon as three heads or three tails in a row has been reached, tossing ends.
DEFDBL A-Z
OPEN "ctsim1.txt" FOR OUTPUT AS #2
RANDOMIZE TIMER
FOR n = 3 TO 12
PRINT #2, n
bothCt = 0: neitherCt = 0: only3TCt = 0: only3HCt = 0
FOR trial = 1 TO 100000
had3T = 0: had3H = 0
headCt = 0: tailCt = 0
FOR toss = 1 TO n
heads = INT(2 * RND(1))
IF heads THEN
headCt = headCt + 1
tailCt = 0
ELSE
headCt = 0
tailCt = tailCt + 1
END IF
t = toss ' hold for later, as NEXT leaves incremented by 1
IF tailCt = 3 THEN had3T = 1: EXIT FOR
IF headCt = 3 THEN had3H = 1: EXIT FOR
NEXT
IF t = n THEN
IF had3T THEN
IF had3H THEN
bothCt = bothCt + 1
ELSE
only3TCt = only3TCt + 1
END IF
ELSE
IF had3H THEN
only3HCt = only3HCt + 1
ELSE
neitherCt = neitherCt + 1
END IF
END IF
END IF
NEXT trial
PRINT #2, " hadTTT noTTT"
PRINT #2, "hadHHH"; : PRINT #2, USING " ####### #######"; bothCt; only3HCt
PRINT #2, " noHHH"; : PRINT #2, USING " ####### #######"; only3TCt; neitherCt
PRINT #2, " "; : PRINT #2, USING " ####### #######"; only3TCt + bothCt; neitherCt + only3HCt
PRINT #2,
NEXT n
CLOSE
The results are
3
hadTTT noTTT
hadHHH 0 12519
noHHH 12487 74994
12487 87513
4
hadTTT noTTT
hadHHH 0 6256
noHHH 6264 62246
6264 68502
5
hadTTT noTTT
hadHHH 0 6174
noHHH 6277 50138
6277 56312
6
hadTTT noTTT
hadHHH 0 4635
noHHH 4707 40765
4707 45400
7
hadTTT noTTT
hadHHH 0 3909
noHHH 4021 32619
4021 36528
8
hadTTT noTTT
hadHHH 0 3203
noHHH 3137 26671
3137 29874
9
hadTTT noTTT
hadHHH 0 2465
noHHH 2609 21382
2609 23847
10
hadTTT noTTT
hadHHH 0 2003
noHHH 2048 17395
2048 19398
11
hadTTT noTTT
hadHHH 0 1670
noHHH 1665 13945
1665 15615
12
hadTTT noTTT
hadHHH 0 1347
noHHH 1405 11445
1405 12792
As expected, when n=3, all trials proceeded to completion of the three tosses, and in 1/8 of the cases TTT was achieved and in another 1/8 of the cases HHH was achieved.
With 100,000 trials, a 50-50 probability of getting the observed results (going all the way to n tosses, and getting no TTT's) would give 50,000 such results. That is most closely matched when t=5, with 56,312 of that set of events happening and when t=6, with 45,400 of the trials reaching 6 tosses with no TTT. In the latter case, of the 45,400 reaching 6 tosses with no TTT, 40,765 also had no HHH, for a conditional prob of 89.79%.
If on the other hand we always allow the tossing to continue until n tosses have been reached, this opens the possibility that you will get both TTT and HHH. The simulation program is the same as the above but without the EXIT FOR's. The results are:
3
hadTTT noTTT
hadHHH 0 12263
noHHH 12437 75300
12437 87563
4
hadTTT noTTT
hadHHH 0 18811
noHHH 18721 62468
18721 81279
5
hadTTT noTTT
hadHHH 0 24835
noHHH 25148 50017
25148 74852
6
hadTTT noTTT
hadHHH 3104 28210
noHHH 27900 40786
31004 68996
7
hadTTT noTTT
hadHHH 6272 30440
noHHH 30680 32608
36952 63048
8
hadTTT noTTT
hadHHH 10163 31680
noHHH 31602 26555
41765 58235
9
hadTTT noTTT
hadHHH 14363 31988
noHHH 32151 21498
46514 53486
10
hadTTT noTTT
hadHHH 18763 31808
noHHH 32097 17332
50860 49140
11
hadTTT noTTT
hadHHH 23481 31569
noHHH 30996 13954
54477 45523
12
hadTTT noTTT
hadHHH 28055 30167
noHHH 30451 11327
58506 41494
In this case, it is n=10 that has the closest to a 50-50 split between hadTTT and noTTT. Since it always goes to completion of the n tosses, all 100,000 are shown. Of the 49,140 that had no TTT in 10 tosses, 17,332 also had no HHH, for a conditional prob. of 35.37%.
|
|
Posted by Charlie
on 2004-06-17 11:12:25 |
simulations
