that there are four red ladybugs, four green lady bugs, and four yellow ladybugs. Upon closer examination I see that one of the red lady bugs has one spot, another has two spots, the third has three spots, and the fourth has four spots. The same is also the case for the green and yellow ladybugs.
All of a sudden, five of these ladybugs fly through my open window and take up residence on my Creeping Charlie. What is the probability that three of them have the same number of spots, and two others also have the same number of spots but a different number of spots than the first three ladybugs have?
My Garden Probability Problem No. 1
My Garden Probability Problem No. 2
While working in my garden, I see seven caterpillars. They're so cute that I would give them pet names, but what's the point? They all look alike. After they moved around for awhile, I wouldn't know who was who anymore. Anyway, these caterpillars might crawl out of my garden and leave. They can do so on the north, east, south, or west side. If all of these caterpillars decide to leave my garden, in how many different ways can they do so?
For example, one possible way for all of the caterpillars to leave my garden would be that one caterpillar leaves by the south side, and the rest of them leave by the north side.
Do you want a hint ?
An explanation of this problem is coming soon. Particularly important is the fact that each caterpillar is identical. We know that the caterpillars are not ultimately identical. If there is a God, at least God knows the difference. But they all look alike to me. How does this affect the problem? Perhaps even I know the difference, but I just dont care. Does that affect the problem? I'll let you know.
My Garden Probability Problem No. 3
After devouring all the pests on my Creeping Charlie, the ladybugs in problem no. 1 fly back to my garden, into which two unlucky aphids have just wandered. One is a male. The other is a female. They don't have a chance with my twelve ladybugs present. My ladybugs are such voracious eaters that it is impossible for any one of them to eat both aphids. The others will be too fast.
What is the probability that the red ladybug with two spots eats the male aphid, and the yellow ladybug with three spots eats the female aphid?
My Garden Probability Problem No. 4
As I ponder cutting my garden up into square plots, I realize that if I make a 2x2 garden (4 square plots), that I can actually count 5 squares -- the big one that is actually the whole garden plus the 4 plots I just made.
If I make a 3x3 garden, I can count 14 squares.
How many squares can I count if I make an 8x8 garden?
What is the general formula for the number of squares I can count for an nxn garden?
My Garden Probability Problem No. 5
After I cultivate my garden into 16 different square plots ( 4x4 ), I see in the distance 16 identical-looking praying mantises heading randomly for my garden. What is the probability that 2 of them will land in the 2nd plot, 4 will land in the 6th plot, and 10 land in the 16th plot?
My Garden Probability Problem No. 6
Referring to the praying mantises in Garden Problem No. 5, what is the probability that 2 of them will land in the same plot, 4 will land in the same plot but not the same plot as the first 2 mantises occupied, and 10 will land in the same plot but not in either of the first 2 occupied plots?
My Garden Probability Problem No. 7
One of my praying mantises has been meditating on a group of 12 different pests. He wants to eat 4 of them for dinner, and while each one is delicious by itself, he notices that the ones that are next to each other make up a horrible taste combination. They are standing in a straight line, but still... He's having one hard time deciding what to do. How many choices does my praying mantis have for eating 4 pests so that his dinner tastes good?
My Garden Probability Problem No. 8
A swarm of White Flies is headed in the general direction of my garden. Somewhere from 1 to 16 White Flies will land in the first plot of my garden with a probability of 1/16 for each possible occurrence. The same is also the case for the rest of the 16 plots in my garden. What is the probability that a total of 33 white flies will land in my garden?
My Garden Probability Problem No. 9
I have decided to collect randomly 4 ladybugs from the 12 that are in my garden ( See prob. no. 1 & 3 ). What is the probability that I will collect at least 1 ladybug of each different color?
My Garden Probability Problem No. 10
After their meditation period, two of my praying mantises decide they would like to play a game. Collecting four ladybugs with one, two, three, and four spots respectively, they put them in a bag and begin drawing them out of the bag one at a time noting the number of spots. They put the ladybug back in the bag, shake it up good for the sake of randomness and draw out another ladybug, noting its number of spots. Now they add up the number of spots. If the sum is 3, the first mantis wins. If the sum is 5, the second mantis wins. If the sum is neither a 3 or a 5, they do the process all over again until one of them wins the bet. After a few games, the first praying mantis begins to wonder if he has a good bet? What is the first mantis's probability of winning?
- Hint: It is theoretically possible for neither mantis to win no matter how many times they select ladybugs.
If you are disappointed that there aren't more problems, don't worry. My garden is full of problems, and I am looking for them every day.
In the meantime,
- Let's think sequence is here (in part).
