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dusya [7]
4 years ago
9

Leslie is a biologist. She is going to randomly select one animal from her lab to study. There are 5 salamanders, 3 crayfish, an

d 12 minnows in her lab. What is text P salamander P salamander start text, P, left parenthesis, salamander end text, right parenthesis? If necessary, round your answer to 2 decimal places. In decimal form
Mathematics
2 answers:
Virty [35]4 years ago
4 0

Answer:

is there answer choices?

Step-by-step explanation:

nika2105 [10]4 years ago
4 0

Answer:

0.25

Step-by-step explanation:

Probability = Number of favorable outcomes / Number of posssible

outcomes

There are 5 favorable outcomes (the 5 salamanders).

There are 20 posssible outcomes (the 5+3+12=20 total animals).

P (salamander) = \frac{5}{20} = 0.25

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Prove that if $w,z$ are complex numbers such that $|w|=|z|=1$ and $wz\ne -1$, then $\frac{w+z}{1+wz}$ is a real number.
mr_godi [17]

Answer:

See proof below

Step-by-step explanation:

Let r=\frac{w+z}{1+wz}. If w=-z, then r=0 and r is real. Suppose that w≠-z, that is, r≠0.

Remember this useful identity: if x is a complex number then x\bar{x}=|x|^2 where \bar{x} is the conjugate of x.

Now, using the properties of the conjugate (the conjugate of the sum(product) of two numbers is the sum(product) of the conjugates):

\frac{r}{\bar{r}}=\frac{w+z}{1+wz} \left(\frac{1+\bar{w}\bar{z}}{\bar{w}+\bar{z}}{\right)

=\frac{(w+z)(1+\bar{w}\bar{z})}{(1+wz)(\bar{w}+\bar{z})}=\frac{w+z+w\bar{w}\bar{z}+z\bar{z}\bar{w}}{\bar{w}+\bar{z}+\bar{w}wz+\bar{z}zw}=\frac{w+z+w+|w|^2\bar{z}+|z|^2\bar{w}}{\bar{w}+\bar{z}+|w|^2z+|z|^2w}=\frac{w+z+\bar{z}+\bar{w}}{\bar{w}+\bar{z}+z+w}=1

Thus \frac{r}{\bar{r}}=1. From this, r=\bar{r}. A complex number is real if and only if it is equal to its conjugate, therefore r is real.

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