STATISTIC (PROBABILITY)

Question

Suppose that using a particular method, there is a probability of 0.85 that a solder joint has a

barrel shape, there is a probability of 0.03 that a solder joint has a cylinder shape, and there is

a probability of 0.12 that a solder joint has an hourglass shape. Furthermore, suppose that

after a certain number of temperature cycles in an accelerated life test, there is a probability of

0.002 that a solder joint is cracked if it has a barrel shape, there is a probability of 0.004 that a

solder joint is cracked if it has a cylinder shape, and there is a probability of 0.005 that a

solder joint is cracked if it has an hourglass shape.

(a) If it is known that a particular joint does not have a barrel shape, determine the

probability that it has a cylinder shape.

(b) If a solder joint is cracked, determine the probability that is has a barrel shape.

(c) Determine the probability that the solder joints are still in good condition after a certain

number of temperature cycles in an accelerated life test.

(d) If a solder joint is not cracked, determine the probability that it has a cylinder shape.

Answer 1

Our first question is pretty straightforward. The basic rule of probability is to look at the number of ways something could occur and divide by the total number of things. So for question 1, we only need to look at the base rates of each of the welds.

Question 1: P(cylinder | ~barrel) = P(cylinder) / (P(cylinder) + P(hourglass))

.03 / (.03 + .12) We substitute the values from our base rate

.03 / .15 Combine the terms in parenthesis

1/5 = .20 or 20% Simplify the fraction

For the other questions, it can be really helpful to make a joint-probabilities table. This allows us to account for Bayesian base rates directly rather than at the equation level.

STEP 1: We can figure out the probability of each type not being cracked by subtracting its given value from 1.

Barrel Cylinder Hourglass

Cracked .002 .004 .005

Notcracked .998 .996 .995

STEP 2: We multiply each of the columns by the by type base rate to figure out the joint percentages

Barrel Cylinder Hourglass

Cracked (.002*.85) (.004*.03) (.005 * .12)

Notcracked (.998*.85) (.996*.03) (.995 * .12)

Barrel Cylinder Hourglass

Cracked .0017 .00012 .0006

Notcracked .8483 .02988 .1194

Step 3: find the sum of the rows in order to figure out the probabilities of being cracked vs not cracked

.0017 + .00012 + .0006 = .00242

.8483 + .02988 + .1194 = .99758

Our final table looks like this

Barrel Cylinder Hourglass Prob of Cracked/NotCracked

Cracked .0017 .00012 .0006 .00242

Notcracked .8483 .02988 .1194 .99758

Prob of Type .85 .03 .12

APPLICATION FOR Q2-3

Now I'll show you how to use this table to solve your problems

Question 2: Probability that weld is barrel shaped if it is cracked. Using the rule we established before, the odds of barrel shaped if cracked is the odds of barrel given cracked / odds of being cracked

P(barrel) = (value on the table where cracked and barrel intersect) / (row total for cracked)

P(barrel) = .0017 / .00242

P(barrel) ≈ .70248 or 70.248%

If you'd prefer the more traditional way of solving this it would look something like this.

P(A|B) / (P(A|B) + P(A|~B))

(.85 * .0002) / ((.85 *.0002) + (.03*.004) + (.12 *.005))

(.0017) / (.0017 + .00012 + .0006)

.0017 / .00242

.70248 or 70.248

Question 3: Probability a weld is not cracked.

Question three is one you already answered in making the joint probability table. It's the sum of the probabilities not cracked row. So you could simply take the row margin of .99758 or 99.758%.

Alternatively you can calculate it as the sum of each base rate times 1 - it's probability of being cracked.

Question 4 is solved using the same method as Question 2. You can choose if you want to use the traditional Bayesian Method, or the Joint Probabilities table.