What is the probability that it will snow on Christmas day in Huntingdon?

Lesson Objective

Understand that some situations are too difficult to model using equally likely outcomes so probabilities need to be found using an alternative technique

Understand how we can estimate the probability of an event using relative frequency

The

Relative

Frequency

of an event

Number of times the event occurs in the experiment

The total number of trials in the experiment

=

Eg I check the weather every day in April.

It rains on 8 of the days, what is the relative frequency

of it raining in April?

Eg Records suggest that the relative frequency of a bus

being late in the morning is 0.1

Over a term of 34 days, on how many days would I

expect the bus to be late?

The Monty Hall Problem

MCPT Mathematics and Technology

About Let’s Make a Deal

• Let’s Make a Deal was a game show hosted by Monty Hall and Carol Merril. It originally ran from 1963 to 1977 on network TV.

• The highlight of the show was the “Big Deal,” where contestants would trade previous winnings for the chance to choose one of three doors and take whatever was behind it--maybe a car, maybe livestock.

• Let’s Make a Deal inspired a probability problem that can confuse and anger the best mathematicians.

Suppose you’re a contestant on Let’s Make a Deal.

You are asked to choose one of three doors.The grand prize is behind one of the doors; The other doors hide silly consolation gifts which Monty called “zonks”.

You choose a door.

Monty, who knows what’s behind each of the doors,reveals a zonk behind one of the other doors.He then gives you the option of switching doors or sticking with your original choice.

You choose a door.

The question is: should you switch?

Monty, who knows what’s behind each of the doors,reveals a zonk behind one of the other doors.He then gives you the option of switching doors or sticking with your original choice.

X X Win

X X Win

X Win X

Win X X

True or Not?

At the start of the game there is a 1/3 chance of me picking the car.

I now know one of the doors which has a zonk behind it, so there are two doors left, one of

which has the car and one of which has a zonk. Therefore the chances of me winning the car is

now 1/2 for either door.

Conclusion: There is no point me changing my door

Is this true?

We are going to simulate the game in pairs.

One player in each pair will be Monty (the host) and the other player will be the contestant

‘The Changers’ will play the game and always change the door they select

‘The Stickers’ will play the game but never change the door they select.

Each pair needs to play the game 10 times and record how many wins

The correct answer:

You should change your choice, because the probability of you winning the car if you do is 2/3.

You pick a door randomly

Pick a door

With a Zonk

Pick a door

With a Zonk

Pick a door

With a Car

Keep

Win a

Zonk

Keep

Win a

Zonk

Keep

Win a

Car

Change

Win a

Car

Change

Win a

Car

Change

Win a

Zonk

Plenary questions:

Why do we need to use relative frequency?

How do you calculate the relative frequency?

Do you think you will get better results by calculating relative frequencies based on 10 experiments or 50 experiments?