Download Probability Lab

PROBABILITY
INTRODUCTION
Take a chance! If 200 chances to win a guitar were sold and you bought 10, what would be the
probability that you would win? Probability is expressed as a ratio: the number of times something
occurs (10) over the total number of possible occurrences (200). So, you can calculate the probability
of winning by dividing 10 by 200. In this case, the probability (p) would be p = 10/200 or .05, or .05 X
100 = 5%. Probability is a numerical expression of the likelihood that an event will occur.
PURPOSE
Write a purpose for this lab on your answer sheet.
PROCEDURE
A. Sample Size
When you toss a coin, what is the probability that it will land with the head side up? The answer is
1/2, or .5, or 50 percent. In 10 throws you would expect to get 5 heads and 5 tails.
When you actually throw the coin, you may not get a 5 heads : 5 tails ratio. The difference between
what you expect according to probability and what happens is called the deviation. For example, if
you get 6 heads and 4 tails, your deviation is 2 (1 from the expected 5 heads plus 1 from the
expected 5 tails).
Deviation is normally expressed as a percent. The percent deviation is calculated as follows:
percent deviation =
X 100
deviation
total number of occurrences
= 2 X 100 = 20 %
10
Answer question 1.
Toss a coin 10 times and record the results in Table 1. Repeat the process for 9 more trials of 10
throws each, making 100 throws in all. Calculate the percent deviation for each trial and then total
the number of heads and tails for the 100 throws and calculate the percent deviation for the 100
throws. Put your data (number of heads and tails) for your 100 throws on the board. Total the
samples of 100 for the entire class from the data on the front board. Answer questions 2-4.
B. The Product Rule
The product rule states the probability of two independent events occurring simultaneously.
Independent events are events that do not influence one another. For example, if you roll two dice,
the number that comes up on one does not influence the number that comes up on the other.
In this section you will use coins to discover the product rule of probability.
Flip 2 coins 100 times. Tally the results in Table 2 and then answer question 5.
The probability of getting heads on the first coin is 1/2, or 0.5 and the probability of getting heads on
the second coin is also 1/2. You can determine the probability that both coins will come up heads by
multiplying the individual probabilities.
1/2 X 1/2 = 1/4 or .25
Note that there are two ways of getting 1 head and 1 tail - the head first or the tail first. You can find
the probability of getting 1 head and 1 tail by adding the two individual probabilities.
First coin
Second coin
head
tail
p = 1/2 X 1/2 = 1/4
tail
head
p = 1/2 X 1/2 = 1/4
or
p = 1/4 + 1/4 = 1/2 or .50
To summarize, the probability of getting the different combinations of the two coins is:
2 heads
p = 1/4 or .25
1 head, 1 tail
p = 1/2 or .50
2 tails
p = 1/4 or .25
Answer questions 6-9.
C. The Chi-square Statistical Method
If you throw a die, what is the probability of a 4 coming up? Because there are 6 sides on the die,
the probability is 1 in 6, or 0.167, or 16.7 percent. In 60 rolls, you would expect the 4 to come up 10
times. If the 4 came up 9 times, you would probably think that the deviation was due to chance.
However, what if the 4 came up 40 times in the 60 throws? Because the deviation is so great, you
might think that the die was “loaded” - that the deviation was not due to chance. The chi-square is a
mathematical tool designed to show the probability that chance alone is operating in a situation.
Chi is a Greek letter that looks like an x. Chi-square is normally written “x2”. It is derived by this
equation:
x2 =
(expected outcome - observed outcome)2
expected outcome
3
In the equation,
means to sum the value obtained for each data category. A “category” refers to
the possible things that can occur. For example, when you flip a coin you can get a head or a tail, so
there are 2 data categories. When you try to roll a 4 on a die, you either roll the 4 or another number
(1, 2, 3, 5, or 6). In this case the 2 categories are “4” and “the other numbers”. (The number of data
categories can be higher, depending on the situation.)
In the die example for 60 throws, we expected 10 fours and 50 other numbers (1, 2, 3, 5, or 6).
We observed 9 fours and 51 others. In using the chi-square, add together the value obtained for the
4-category and the value obtained for the other number-category.
x2 = (10 - 9)2 + (50 - 51)2
10
50
=
12
+ - 12
10
50
=
1
+
1
10
50
=
.1
+ .02
2
x =
.12
Once a chi-square value is determined it must be interpreted with the use of the chi-square table
which is found on page 4.
Because there are 2 data categories, look for the chi-square value on the line marked “For 2 data
categories”. The chi-square value of .12 falls between the values listed in column 2 (.004) and
column 3 (.455).
The percent probability that chance alone is operating - the p-value - is shown at the bottom of each
column. In column 2 the figure is 95 percent and in column 3 it is 50 percent. So the chi-square
value of .12 means that the probability is between 50 percent and 95 percent that chance alone will
account for this much deviation. In other words, if 4 comes up 9 times in 60 rolls, the deviation is
likely due to chance.
If the p-value turns out to be less than 5 percent (chi-square value higher than 3.841), something
other than chance is thought to be involved. In the example of 4 coming up 40 times in 60 rolls, the
chi-square value is 108. One would suspect that the die was loaded. Answer questions 10-12
Chi-square Values
For 2 data
categories
For 3 data
categories
For 4 data
categories
Probability that
chance alone is
responsible for the
deviation
.001
.004
.455
1.074
1.642
2.706
3.841
6.635
.020
.103
1.386
2.408
3.219
4.605
5.991
9.210
.115
.352
2.366
3.665
4.642
6.251
7.815 11.341
99%
95%
50%
30%
20%
10%
5%
1%
PROBABILITY LAB
NAME _________________________
DATE ______________ PERIOD ___
Write a purpose for this lab. _______________________________________________________
_____________________________________________________________________________
1. Calculate the percent deviation for 3 heads and 7 tails out of 10 throws. (show work)
Table 1
TRIAL(10
throws each)
NUMBER OF
HEADS
NUMBER OF
TAILS
PERCENT
DEVIATION
1
2
3
4
5
6
7
8
9
10
TOTALS
___________
____________
PERCENT DEVIATION FOR 100 THROWS
(show work)
_____________
2. How does the deviation for a sample size of 10 throws (trials 1-10) compare to the deviation for a
sample size of 100 throws?
__________________________________________________________________________
3. On line “a” calculate to the nearest 0.1 the percent deviation for the class sample. (show work)
On line “b” explain how the deviation for the class sample compares to the deviation for your
sample of 100 throws?
a.Heads
b
Tails
4. Look at your answers to questions 2 and 3 and explain the relationship between sample size,
deviation and probability.
__________________________________________________________________________
__________________________________________________________________________
Table 2
2 HEADS
1 TAIL, 1 HEAD
2 TAILS
TOTALS
RATIOS*
*Ratios are calculated by dividing all the numbers by the smallest number. Calculate each number to
the nearest .1 above.
5. What is the nearest whole-number ratio that will best compare the results in Table 2? (1:1:1,
2:1:1, etc)
_____________________
6. What is the probability of getting 2 tails on two different coins on the same throw? (show work)
__________________________________________
7. What is the probability of getting a head on the first coin and a tail on the second coin on the same
throw? (show work)
__________________________________________
8. Refer to the data in Table 2 for your 100 throws of the two coins and calculate the percent
deviation from the expected probability. First calculate the number of occurrences of each
combination expected in 100 throws. Then calculate the deviation between your results and the
number of actual occurrences of each combination.
% DEV= (E-O)+ (E-O)+(E-O) X 100
Total
2 heads
EXPECTED NUMBER OF OCCURRENCES IN 100
THROWS
OBSERVED NUMBER OF OCCURRENCES IN 100 THROWS
PERCENT DEVIATION = ______________ (to nearest .1)
(show work)
1 head
1 tail
2 tails
9. Complete the following statement:
The Product Rule of Probability states that, “To calculate the probability that two independent
events will occur simultaneously, you must......” (look in notes or in lab directions)
__________________________________________________________________________
10. Calculate the Chi-square value and determine the p-value for the following problem.
(SHOW ALL WORK)
In 100 throws of a coin, you get 45 heads and 55 tails.
p-value ____________
11. This problem has three data categories. The chi-square value you should get is 2.92. When
you get the correct chi-square value, determine the p-value. (SHOW ALL WORK)
In 200 throws of two coins, you get 45 heads, 112 heads/tails combinations, and 43 tails.
p-value ____________
12. A single coin is tossed 50 times. Heads come up 20 times and tails 30 times. What is the
probability that chance alone is responsible for this deviation? (SHOW ALL WORK)
p-value ____________