Question

A coin is flipped. if a heads is flipped, then the coin is flipped 4 more times and the number of heads flipped is noted; otherwise (i.e., a tails is flipped on the initial flip), then the coin is flipped 2 more times and the result of each flip (i.e., heads or tails) is noted successively. how many possible outcomes are in the sample space of this experiment?

Answer 1

The first flip can obviously return only H or T.

Suppose the coin lands on H. This means that every following string is possible (and thus part of the sample space)

$x_1,\ x_2,\ x_3\ x_4,\quad x_i \in \{H,\ T\}$

So, there are 16 such strings, because every flip has two possible outcomes.

Suppose now the coin lands on T. Like before, we add to the sample space the following strings:

$x_1,\ x_2,\quad x_i \in \{H,\ T\}$

And thus there are 4 such strings, because we have two choices for each of the two flips.

So, the sample space has 20 outcomes:

H - HHHH

H - HHHT

H - HHTH

H - HHTT

H - HTHH

H - HTHT

H - HTTH

H - HTTT

H - THHH

H - THHT

H - THTH

H - THTT

H - TTHH

H - TTHT

H - TTTH

H - TTTT

T - HH

T - HT

T - TH

T - TT