If two events or states of affairs are independent, then the probability that both will occur is equal to the multiplication of the probabilities of those two events.
If p is an event (or state of affairs) that is independent of an event (or state of affairs) q, then:
P(p & q) = P(p) x P(q)
But if p and q are dependent events, then the probability formula is a bit different:
P(p & q) = P(p) x P(q/p)
Suppose p is ‘getting heads on coin toss 1’ and q is ‘getting heads on coin toss 2’. Assuming that the outcome of coin toss 1 has no influence on the outcome of coin toss 2, we can conclude that p and q are independent events, and use the first, simpler formula above:
P(p & q) = .5 x .5 = .25
But if p is “It will rain in Seattle today” and q is “The streets in Seattle will get wet today”, then the truth or falsehood of p has an obvious influence on the truth or falsehood of q, namely if p is true, then it is virtually certain that q will also be true. Since these are dependent events, we must use the second, more complex formula to calculate the probability of the conjunction of the two claims:
P(p & q) = P(p) x P(q/p)
Since it is virtually certain that q is the case given the assumption that p is the case, P(q/p) is approximately equal to 1.0, so the probability that both p and q are true is about the same as the probability that p is true.
In arguments where two or three premises work together to support the conclusion, each of the two or three premises must be true in order for the argument to work and provide rational support for the conclusion. We can thus determine the probability that all two or three premises are true by using one of the above probability formulas. But we should use the simpler probability formula (where the probability of each event/premise is multiplied with the probability of the other premise(s) only if the events/premises are independent.
One example of mistaken multiplication of probabilities occurs in arguments about Jesus allegedly fulfilling Old Testament prophecies:
Stoner [Peter Stoner] says [in Science Speaks] that by using the modern science of probability in reference to eight prophecies…”We find that the chance that any man might have lived down to the present time and fulfilled all eight prophecies is 1 in [10 to the 17th power].” That would be 1 in 100,000,000,000,000,000.(Josh McDowell, Evidence that Demands a Verdict, revised edition, p.167)
McDowell does not bother to provide Stoner’s calculations, but I strongly suspect that the probability of each of the eight predictions was multiplied together to get the tiny probability mentioned above. If so, then Stoner mistakenly applied the simple formula for determining probability of a conjunction of events. The problem with using the simple formula is that it assumes that all of the events are independent from each other, but this is not the case with the eight prophecies:
1.The messiah will be born at Bethlehem (based on Micah 5:2).
2. The messiah will be preceded by a messenger (based on Malachi 3:1).
3. The messiah will enter Jerusalem on a donkey (based on Zechariah 9:9).
4. The messiah will be betrayed by a friend (based on Psalms 41:9).
5. The messiah will be sold for 30 pieces of silver (based on Zechariah 11:12)
6. The 30 pieces of silver [received from the ‘sale’ of the messiah] will be thrown in the house of God and then used to buy a potter’s field (based on Zechariah 11:13)
7. The messiah will be silent before his accusers (based on Isaiah 53:7)
8. The messiah’s hands and feet will be pierced when he is executed along with criminals (based on Psalm 22:16 and Isaiah 53:12).
Obviously, if (6) is true, then (5) must also be true, because (6) presupposes the truth of (5). So, if the probability of (5) applying to a randomly chosen person was, say, one chance in a billion, it would be a mistake to multiply that probability times the probability of (6) in order to get the probability that both (5) and (6) were the case. Rather, the probability of both (5) and (6) being true is simply the probability of (6) being true, because if (6) were true, then (5) would automatically and necessarily also be true.
What is that chance that some randomly chosen human being will (in his/her lifetime) enter Jerusalem riding a donkey? Out of the billions of people who are alive right now, only about 800,000 live in Jerusalem. But lots of people visit Jerusalem, so perhaps millions of people will enter Jerusalem each year. Only a small fraction of the visitors will ride donkeys as they enter the city. Most will ride a car, bus, truck, motorcycle, or bicycle. In a decade, I would guess that no more than a million people ride into Jerusalem on a donkey (in modern times). In a century, no more than 10 million would do so. In one century perhaps 10 billion people will be born and live, so a rough ratio would be 10 million out of 10 billion, or one million out of one billion, or one out of a thousand. That seems a bit high, but let’s just go with that rough estimate for purposes of illustration.
We cannot take the small probability that a randomly chosen person would enter Jerusalem on a donkey (.001) and simply multiply that times the probability that a randomly chosen person would have been born in Bethlehem in order to arrive at the probability that some random person would satisfy both (1) and (3). A person who was born in Bethlehem has a much greater chance of riding a donkey into Jerusalem than just a randomly selected human being. Such a person might well have one chance in a hundred of riding a donkey into Jerusalem (probability =.01) or possibly even one chance in twenty of doing so (probability =.05).
Finally, there is another obvious relationship between (5) and (4). Jesus was not literally ‘sold’ for thirty pieces of silver. He was ‘sold out’ for thirty pieces of silver (or some amount of money). He was, that is, betrayed for a sum of money. If Jesus was betrayed for a sum of money, that makes it somewhat likely that he was betrayed by a friend, for a friend is often in a position to betray one, while strangers and others are not so often in such a position. In any case, the truth of (5) makes it probable (or certain?) that Jesus was betrayed, which is part of the way towards (4) being true, thus (5) increases the likelihood of (4). These are not two independent events.
Side note: In my view some of these eight predictions are false (or probably false). Nails were probably not used in Jesus’ crucifixion, in which case Jesus’ hands and feet were not pierced. If nails were used, they might have only been used on his feet (such as with the one and only example of the bones of a crucified man found in Palestine), or only used on his hands and not his feet. Furthermore, if nails were used to attach Jesus’ arms to the cross, most scholars believe that he would have been nailed through the wrists, in which case his hands would NOT have been pierced. In my view Jesus was probably not born in Bethlehem, but rather in Nazareth. Many NT scholars doubt the historicity of the birth stories found in Matthew and Luke. Although Judas might have accepted a bribe to betray Jesus, I doubt that anyone other than Judas and the person who gave the bribe knew how much money and how many silver coins were involved. In that case it is very unlikely that the bribe was exactly 30 pieces of silver. The gospels themselves clearly indicate that Jesus spoke when he was accused, both in the (alleged) Jewish trial, as well as when he was (allegedly) tried by Pilate, so the claim that Jesus was silent before his accusers is false (or probably false). If Jesus did ride into Jerusalem on a donkey, there is a good chance that he did so precisely in order to satisfy prediction (3), which would nullify whatever assumed tiny probability was assigned to that prediction by Peter Stoner in his calculations.=================
So, the point of this discussion is simply to highlight an important qualification in the use of multiplying probabilities of premises: If the event or state of affairs of one of the premises has some causal or logical relation to an event or state of affairs asserted in one of the other premises, such that the truth of one premise would have a significant impact on the probability of the other premise, then one cannot simply multiply the probabilities of the premises in order to determine the probability that all of the premises are true.