g If the quantity of helium in a weather balloon in increased by a factor of 5 at constant temperature and pressure, by how much does the volume of the balloon change? Decrease by 2 times Increase by 5 times Decrease by 5 times Increase by 2 times

Increase by 5 times

Step-by-step explanation:

Given that from the ideal gas law;

PV = nRT

P= pressure

V= volume

n= number of moles

R= gas constant

T = temperature

It the follows that;

V= nRT/P

When the number of moles (quantity of helium) is increased by a factor of 5 while temperature and pressure remain constant, we can write;

PV = 5nRT

V = 5nRT/P

Hence the volume increases by a factor of 5.

a) p⁴

b) ( 1 - p)p³

c) 1 - p⁴

Step-by-step explanation:

Data provided in the question:

The coin flips are independent

Probability that coin lands on head, P(H) = p

Thus,

Probability that coin lands on tails, P(T) = ( 1 - p )

Now,

a) P( H,H,H,H ) = P(H) × P(H) × P(H) × P(H)

or

⇒ P( H,H,H,H ) = p × p × p × p = p⁴

b) P( T,H,H,H ) = P(T) × P(H) × P(H) × P(H)

or

⇒ P( T,H,H,H ) = ( 1 - p) × p × p × p = ( 1 - p)p³

c) probability that the pattern T, H, H, H occurs before the pattern H, H, H, H

For obtaining four consecutive Heads (H) we need to get 3 consecutive Heads first.

let we have found our first sequence somewhere in our sequence. Now, observing the throw before the first H. So we can have a Tails (T), i.e sequence THHH will appear first, or we can have no throw because the first H was our first throw.

(It can't be a H otherwise your sequence would not be the first one to occur.)

Thus, the only way for to come first is to start with it, and P() = p⁴.

Hence,

P( T, H, H, H occurs before the pattern H, H, H, H) = 1 - p⁴