Solve the given problem that involves Probability

In summary: Actually, I didn't omit...I just indicated the sum total from the first to last term... would yield same result. Apologies for confusion.
  • #1
chwala
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Homework Statement
See attached;
Relevant Equations
Probability
I may seek an alternative approach; actually i had thought that this would take a few minutes of my time..but just realized that it just takes a minute; My interest is only on highlighted part.

1677496127148.png


Text solution

1677496186076.png


My take;

##P(\text{at least one of the first three days is wet})=1-P(ddd)##
=## 1-(0.6×0.7×0.7)=0.706##

Of course the other way of doing it would also realize the same result but will need more time...i.e using
##P(www)+P(wwd)+P(wdw)+P(wdd)+P(dww)+...P(ddw)=0.706##

Cheers man!
 
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  • #2
chwala said:
My take;
##P(\text{at least one of the first three days is wet})=1-P(ddd)##
=## 1-(0.6×0.7×0.7)=0.706##
IMO, the approach above is the better approach. It uses the idea that the complement of "at least one day of the three is wet" is "none of the three days is wet." The probabilities of these two events has to add to 1.
chwala said:
Of course the other way of doing it would also realize the same result but will need more time...i.e using
##P(www)+P(wwd)+P(wdw)+P(wdd)+P(dww)+...P(ddw)=0.706##
The latter approach is trickier to get right in that you have to ensure that you have included all possible events in which one, two, or three of the days is wet. In the probabilities that you list, you have omitted one. These come from the following combinations.
3 wet days: ##\binom {3}{3} = 1## -- www
2 wet days: ##\binom{3}{2} = 3## -- wwd, wdw, dww
1 wet day: ##\binom{3}{1} - 3## -- wdd, dwd, ddw
 
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  • #3
Mark44 said:
IMO, the approach above is the better approach. It uses the idea that the complement of "at least one day of the three is wet" is "none of the three days is wet." The probabilities of these two events has to add to 1.
The latter approach is trickier to get right in that you have to ensure that you have included all possible events in which one, two, or three of the days is wet. In the probabilities that you list, you have omitted one. These come from the following combinations.
3 wet days: ##\binom {3}{3} = 1## -- www
2 wet days: ##\binom{3}{2} = 3## -- wwd, wdw, dww
1 wet day: ##\binom{3}{1} - 3## -- wdd, dwd, ddw
Actually, I didn't omit...I just indicated the sum total from the first to last term... would yield same result. Apologies for confusion.
 

What is probability and why is it important?

Probability is a measure of the likelihood that an event will occur. It is important because it allows us to make predictions and decisions based on the chances of different outcomes.

How do you calculate probability?

Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. This gives a decimal or fraction that can be converted to a percentage to represent the likelihood of the event occurring.

What is the difference between theoretical and experimental probability?

Theoretical probability is based on mathematical calculations and assumes that all outcomes are equally likely. Experimental probability is based on actual results from experiments or observations and may differ from theoretical probability due to chance or other factors.

How can probability be used in real life situations?

Probability can be used in many real life situations, such as predicting the weather, determining the chances of winning a game or lottery, and making decisions in business or finance. It can also be used to analyze risk and make informed choices.

What are some common misconceptions about probability?

Some common misconceptions about probability include the belief that past outcomes can influence future outcomes, that rare events are more likely to occur, and that independent events are somehow connected. It is important to understand the principles of probability in order to avoid these misconceptions.

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